Research Article Direct Scaling Analysis of Fermionic Multiparticle … · 2019. 7. 30. · obtained for a xed energy (cf. [, ]) or in an energy interval (cf. [, ]) into the language

Research ArticleDirect Scaling Analysis of Fermionic Multiparticle CorrelatedAnderson Models with Infinite-Range Interaction

Victor Chulaevsky

Departement de Mathematiques Universite de Reims Moulin de la Housse BP 1039 51687 Reims Cedex 2 France

Correspondence should be addressed to Victor Chulaevsky pliniusfreefr

Received 26 August 2015 Accepted 13 December 2015

Academic Editor Hagen Neidhardt

Copyright copy 2016 Victor Chulaevsky This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

We adapt themethod of direct scaling analysis developed earlier for single-particle Andersonmodels to the fermionicmultiparticlemodels with finite or infinite interaction on graphs Combined with a recent eigenvalue concentration bound for multiparticlesystems the new method leads to a simpler proof of the multiparticle dynamical localization with optimal decay bounds in anatural distance in the multiparticle configuration space for a large class of strongly mixing random external potentials Earlierresults required the random potential to be IID

1 Introduction

11 The Model A Brief Historical Overview Analysis oflocalization phenomena in multiparticle quantum systemswith nontrivial interaction in a random environment is arelatively new direction in the Anderson localization theorywhere during almost half a century since the seminal paperby Anderson [1] most efforts were concentrated on the studyof disordered systems in the single-particle approximationthat is without interparticle interaction While in numer-ous physical models such an approximation seems fairlyreasonable it was pointed out already in the first worksby Anderson that the multiparticle models presented a realchallenge

The number of results on multiparticle localizationobtained in the physical and mathematical communitiesstill remains rather limited We do not review here resultsobtained by physicists (cf [2 3]) based on the methods oftheoretical physics and considered as firmly established bythe physical community The rigorous mathematical resultson multiparticle localization obtained so far (cf eg [4ndash7])apply to 119873-particle systems with arbitrary but fixed 119873 gt 1and the range of parameters (such as the amplitude of thedisorder andor the proximity to the edge(s) of the spectrum)is rapidly degrading as119873 rarr infin

In this paper we study spectral properties of the multi-particle random Hamiltonians of the form

H (120596) = H0+ 119892V (120596) + U 119892 isin R (1)

describing 119873 gt 1 fermionic particles in the configurationspaceZ assumed to be a finite or countable graph For claritywe present first our method in the particular case whereZ = Z1 and then describe how to adapt the argumentsto more general graphs Z with polynomial growth of balls(cf Section 8) In (1) H

0is a finite-difference operator

representing the kinetic energy for example the nearest-neighbor discrete negative Laplacian (minusΔ) U is the operatorof multiplication by the interaction potential x 997891rarr U(x) andV(120596) is the operator ofmultiplication by the random function

x 997891997888rarr V (x 120596) = 119881 (1199091 120596) + sdot sdot sdot + 119881 (119909

119873 120596) (2)

where 119881 Z times Ω rarr R is a random field relative to someprobability space (ΩFP) 119892 gt 0 is a parameter measuringthe amplitude of disorder (All notations are explained indetail in Section 2)

We consider only the restriction ofH(120596) to the fermionicsubspace The bosonic subspace can be treated essentially inthe same way

Hindawi Publishing CorporationAdvances in Mathematical PhysicsVolume 2016 Article ID 2129682 17 pageshttpdxdoiorg10115520162129682

2 Advances in Mathematical Physics

12Motivation forThisWork The twomost popularmethodsof the rigorous Anderson localization theory the MultiscaleAnalysis developed in [8ndash11] and the Fractional MomentsMethod developed in [12 13] start with the analysis of theGreen functions (GFs) of a given random operator andthen one has to translate the upper bounds on the GFsobtained for a fixed energy (cf [8 11ndash13]) or in an energyinterval (cf [8 10]) into the language of the eigenfunctions(EFs) and eigenfunction correlators (EFCs) In a prior work[14] we proposed an alternative approach called the DirectScaling Analysis of eigenfunctions which focuses essentiallyon the eigenfunctions in finite domains of the configurationspace from the very beginning It keeps some importantadvantages of the MSA namely a greater tolerance than inFMM to lower regularity of the disorder and is close in spiritto the KAM (Kolmogorov-Arnold-Moser) approach whichoften requires a tedious inductive scheme In fact work [14]has been inspired by the KAM approach to quasi-periodicHamiltonians proposed earlier by Sinai [15] Compared to theKAM method the DSA has a simpler structure of the scaleinduction similar to that of theMSA although it also inheritssomewhat weaker probabilistic bounds typical of the MSA

Furthermore as was said there are two kinds of theMSAwith a fixed energy (FEMSA) and with a variable energy(VEMSA) the latter is often referenced to as the energy-interval MSA for it treats the decay properties of the Greenfunctions in an entire interval of energies at onceTheFEMSAis slightly simpler than VEMSA in the single-particle casebut in the realm of multiparticle random Hamiltonians witha nontrivial interaction the variable-energy variant of the(multiparticle) MSA is considerably more complex Some ofthe components of the VEMPMSA are simply eliminatedwith the (MP)DSA approach and the counterparts of someother components become quite elementary A good exampleof such kind is Lemma 21 (see Section 43) the proof ofwhich is very simple and fits in a few lines while its analogin the variable-energy MPMSA (cf eg [4 Lemma 3]) issubstantially more involved

This constatation has been the principal motivation forthe present work A reader familiar with the techniques ofthe papers [4 16 17] can see that the multiparticle variantof the DSA (MPDSA) from the present paper is much closerin its logical structure to the FEMSA yet it leads directly tothe VEMSA-type decay bounds on the entire eigenbases inarbitrarily large finite domains

In preprint [18] which has been a starting point for thepresent paper we gave the first proof of exponential decay ofmultiparticle eigenfunctions in presence of an infinite-rangeinteraction decaying at infinity at a fractional-exponentialrate 119903 997891rarr eminus119903

120577

0 lt 120577 lt 1 In a recent work [7] Aizenmanand Warzel further developing the techniques of [6] provedfractional-exponential decay of EF correlators (EFCs) forsubexponentially decaying interactions of infinite range Dueto the logical structure of the FMM the decay analysis ofthe eigenfunctions is subordinate to that of the EFCs andthis explains why [7] established only a subexponential decayof eigenfunctions The DSA is free from this limitation andthis allows us to prove a genuine exponential decay of the

multiparticle EFs even in the case where the interaction decayis subexponential

13 Novelty Apart from simplifications of the MultiparticleMultiscale Analysis (MPMSA) developed in [4] the noveltyof this paper is that the external random potential is notnecessarily IID (with independent and identically distributedvalues) but is strongly mixing Earlier publications including[19] required the common external random potential actingon the particles to be IID

14 Structure of the Paper The structure of the paper is asfollows

(i) The principal assumptions andmain results are statedin Sections 24ndash26

(ii) The deterministic techniques of scaling analysis arepresented in Section 3

(iii) In Section 4 we treat first the case of finite-rangeinteractions in order to present the logical structureand the methodological advantages of the MPDSA inthe most transparent way

(iv) In Section 7 our analysis is adapted to the interac-tions of infinite range

(v) Sections 5 and 6 are devoted to the derivation ofspectral and dynamical localization from the finalresults of the MPDSA

2 Preliminaries Assumptionsand Main Results

21 Configurations of Indistinguishable Particles in Z1 In thefirst part of this paper we work with configurations of 119873 ge1 quantum particles in the one-dimensional lattice Z =

Z1 In quantum mechanics particles of the same kind areconsidered indistinguishable more precisely depending onthe nature of the particles the wave functions describing119873 gt 1 particles have to be either symmetric (Bose-Einsteinquantum statistics) or antisymmetric (Fermi-Dirac quantumstatistics) We choose here the fermionic case this gives riseto slightly simpler notations and constructions

For clarity we use often boldface symbols for the objectsrelated to the multiparticle systems

Quantum states of an 119873-particle fermionic system in Z

are elements of the Hilbert spaceHN=H119873minus formed by all

square-summable antisymmetric functions Ψ Z119873rarr C

with the inner product ⟨sdot | sdot⟩ inherited from the Hilbertspace ℓ2(Z119873

) = (H1)otimes119873 In this particular case where

the ldquophysicalrdquo configuration space is one-dimensional H119873

admits a simple representation which we will useFirst note that any antisymmetric function

x = (1199091 119909

119873) 997891997888rarr Ψ (119909

1 119909

119873) (3)

vanishes on all hyperplanes x 119909119894= 119909

119895 1 le 119894 lt 119895 le 119873

Further any function defined on the ldquopositive sectorrdquoZNgt=

(1199091 119909

119873) 119909

1gt 119909

2gt sdot sdot sdot gt 119909

119873 admits a unique

Advances in Mathematical Physics 3

antisymmetric continuation to ZN= Z119873 An orthonormal

basis inHN can be chosen in the usual form

Φa (x) =1

radic119873

sum

120587isinS119873

(minus1)120587

119873

⨂

119895=1

1119886120587minus1(119895) (4)

where a isinZN with 1198861 119886

119873 = 119873 and 1

119886is the indicator

function of the single-point set 119886 HereS119873is the symmetric

group acting inZ119873 by permutations of the particle positions120587(a) = (119886

120587minus1(1) 119886

120587minus1(119873)) and (minus1)120587 isin +1 minus1 denotes

the parity of the permutation 120587 It is readily seen thatHN isunitarily isomorphic to the Hilbert space ℓ2(ZN

gt) of square-

summable functions on ZNgt equivalently one can consider

square-summable functions on the setZNge= 119909

1ge sdot sdot sdot ge 119909

119873

vanishing on the boundary ZN== x isin ZN

ge exist119894 = 119895 119909

119894=

119909119895 Indeed the isomorphism is induced by the bijection

between the orthonormal bases Φa and 1a

Φa larrrarr 1a = 11198861otimes sdot sdot sdot otimes 1

119886119873 a isinZN

gt (5)

The subspace HN is invariant under any operator commut-ing with the action of the symmetric groupS

119873

An advantage of the above representation is that ZNgt

inherits its explicit natural graph structure from Z119873 Formore general graphs Z one has to resort to the generalconstruction of the so-called symmetric powers of graphs seethe details in Section 8

22 Fermionic Laplacians Any unordered finite or countableconnected graph (GE) with the set of vertices G and theset of edges E is endowed with the canonical graph distance(119909 119910) 997891rarr dG(119909 119910) (defined as the length 119891 the shortestpath 119909 999492999484 119910 over the edges) and with the canonical graphLaplacian ΔZ

(ΔG119891) (119909) = sum

⟨119909119910⟩

(119891 (119910) minus 119891 (119909))

= minus119899G (119909) 119891 (119909) + sum

⟨119909119910⟩

119891 (119910)

(6)

here ⟨119909 119910⟩ denotes a pair of nearest neighbors and 119899G(119909)is the coordination number of the point 119909 in G that is thenumber of its nearest neighbors

In particular one can take G = Z119873 (or a subgraphthereof) with the edges ⟨119909 119910⟩ formed by vertices 119909 119910 with|119909 minus 119910|

1= |119909

1minus 119910

1| + sdot sdot sdot + |119909

119873minus 119910

119873| = 1 In other words

the vector norm | sdot |1induces the graph distance on Z119873 The

Laplacian on Z119873 which we will now denote by Δ can bewritten as follows

Δ =

119873

sum

119895=1

(

119895minus1

⨂

119894=1

1(119894)) otimes Δ(119895) otimes (119873

⨂

119896=119895+1

1(119896)) (7)

where 1(119894) is the identity operator acting on the 119894th variableand Δ(119895) is the one-dimensional negative lattice Laplacian inthe 119895th variable

(Δ(119895)119891) (119909

119895) = minus2119891 (119909

119895) + 119891 (119909

119895minus 1) + 119891 (119909

119895+ 1)

119909119895isin Z

(8)

The restriction to the subspace HN of antisymmetric func-tions can be equivalently defined in terms of functionssupported by the positive sector ZN

gt hence vanishing on its

borderZN= Indeed the matrix elements of Δ in the basisΦa

can be nonzero only for pairsΦaΦb with |aminusb|1 = 1 so thatfor some 119895 isin [1119873] |119886

119895minus 119887

119895| = 1 while for all 119894 = 119895 119886

119894= 119887

119894

If a b isin ZNgt then ⟨Φa|Δ|Φb⟩ = (1119873)⟨1a|Δ|1b⟩ If say

a isinZN= then the respective matrix element of the Laplacianrsquos

restriction to ZNgt

with Dirichlet boundary conditions onZN

=vanishes and so does the function Φa (which is no

longer an element of the basis in HN) Therefore up to aconstant factor the restriction of the 119873-particle Laplacianto the fermionic subspace HN is unitarily equivalent to thestandard graph Laplacian onZN

gt From this point on we will

work with the latterIt will be convenient in the course of the scaling analysis to

use a different kind of metric onZNgtsub Z119873 the max-distance

defined by

120588 (x y) = max1le119895le119873

dZ (119909119895 119910119895) (9)

and to work with the balls relative to the max-distance

B119871(x) = y 120588 (x y) le 119871 =

119873

⨉

119895=1

119861119871(119909

119895) (10)

Here 119861119871(119909) = [119909 minus 119871 119909 + 119871] cap Z

In the positive sector ZNgt the above factorization of 120588-

balls is subject to the condition that the RHS of (10) is itself asubset of the positive sectorZN

gt(but the inclusion ldquosubrdquo always

holds true) Considering configurations x = 1199091 119909

119899

as subsets of Z one can define the distance between twoconfigurations x1015840 isinZ(n1015840)

gtand x10158401015840 isinZ(n10158401015840)

gtin a usual way

dist (x1015840 x10158401015840) = min119906isinx1015840

minVisinx10158401015840|119906 minus V| (11)

Lemma 1 Let x isin ZNgtbe a union of two subconfigurations

x1015840 isin Z(n1015840)gt

and x10158401015840 isin Z(n10158401015840)gt

such that dist(x1015840 x10158401015840) gt 2119871 Thenthe following identity holds true

B(119873)

119871(x) = B(119899

1015840)

119871(x1015840) times B(119899

10158401015840)

119871(x10158401015840) (12)

The proof is straightforward and will be omittedThe graph distance dZN

gtwill be useful in some geomet-

rical constructions and definitions referring to the graphstructure inherited fromZN

Given a subgraph Λ sub ZNgt we define its internal exter-

nal and the so-called graph (or edge) boundary in terms


of the canonical graph distance (below Λ119888 stands for thecomplement of Λ)

120597minusΛ = y isin Λ dZN

gt(yΛ119888) = 1

120597+Λ = 120597

minusΛ

119888

120597Λ = (x y) isin Λ times Λ119888 dZNgt(x y) = 1

(13)

We also define the occupation numbers of a site 119910 inthe 1-particle configuration space relative to a configurationx = 119909

1 119909

119873 Namely define a function nx Z 997891rarr N by

nx(119910) = 119895 119909119895= 119910 119910 isin X

23 Multiparticle Fermionic Hamiltonians Given a finitesubsetΛ subZN

gt letΣ(H

Λ) be the spectrumof the operatorH

Λ

andGΛ(119864) = (H

Λminus119864)

minus1 its resolventThematrix elements ofresolvents for119864 notin Σ(H

Λ) in the canonical delta-basis usually

referred to as theGreen functions are denoted byGΛ(x y 119864)

In the context of random operators the dependence uponthe element 120596 isin Ω will be often omitted for brevity unlessrequired or instructive Similar notations will be used forinfinite Λ

Given a real-valued functionW ZNgtrarr R we identify

it with the operator of multiplication by W and consider thefermionic119873-particle random Hamiltonian

H (120596) = H0+W (14)

where H0is a second-order finite-difference operator for

example the negative graph Laplacian (minusΔ) Furthermore inour model the potential energy is random

W (x) =W (x 120596) = 119892V (x 120596) + U (x) (15)where 119892V(sdot 120596) is the external random potential energy of theform

119892V (x 120596) = 119892119881 (1199091 120596) + sdot sdot sdot + 119892119881 (119909

119873 120596) (16)

of amplitude 119892 gt 0 and119881 Z timesΩ rarr R is a random field onZ relative to a probability space (ΩFP) The expectationrelative to the measure P will be denoted by E[sdot] Ourassumptions on 119881 are listed below (cf (W1)ndash(W3)) See alsoRemark 4where how (W1)ndash(W3) can be relaxed is explained

Further U is the interaction energy operator for nota-tional brevity we assume that it is generated by a two-bodyinteraction potential 119880

U (x) = sum119894 =119895

119880(10038161003816100381610038161003816119909119894minus 119909

119895

10038161003816100381610038161003816) (17)

satisfying one of hypotheses (U0)-(U1) (see below)Note that without loss of generality if H

0 lt infin

and 119881 is as bounded then the random operator has asbounded norm hence its spectrum is as covered by somebounded interval 119868 sub R In Remark 3 we explain thatthe boundedness of the potential is not crucial for thespectral localization although the statement of the resulton the dynamical localization is to be slightly modified forunbounded potentials

One can also consider higher-order finite-differenceoperators H

0 this requires only minor technical modifica-

tions

24 Assumptions on the Random Potential We formulate theassumptions in the general situation where the 1-particleconfiguration space is a graphZwith graph-distance dZ thisincludes the caseZ = Z1

We assume that the random field 119881 Z times Ω rarr R

is (possibly) correlated but strongly mixing this includes ofcourse the IID potentials treated earlier in [4 6] Let

119865119881119909(119905) = P 119881 (119909 120596) le 119905 119909 isinZ (18)

be the marginal probability distribution functions (PDF) ofthe random field 119881 and

119865119881119909(119905 | F

=119909) = P 119881 (119909 120596) le 119905 | F

=119909 (19)

the conditional distribution functions (CDF) of the randomfield 119881 given the sigma-algebra F119881

=119909generated by random

variables 119881(119910 120596) 119910 = 119909 Our assumptions on correlatedpotentials are as follows

(W1) The marginal CDFs are uniformly Lipschitz continu-ous for some 119862 isin (0 +infin) and all 119904 isin (0 1]

sup119909isinZ

sup119886isinR

[119865119881119909(119886 + 119904 | F

119881

=119909) minus 119865

119881119909(119886 | F

119881

=119909)] le 119862119904 (20)

The LHS of (20) is a conditional probability hence arandom variable so (20) holds P-as

To formulate the next assumption introduce the follow-ing notation given a subset Λ sub Z we denote by F119881

Λthe

sigma-algebra generated by the values of the random poten-tial 119881(119909 120596) 119909 isin Λ

(W2) (Rosenblatt strong mixing) For any pair of subsets 119861101584011986110158401015840sub Z with dZ(119861

1015840 119861

10158401015840) ge 119871 ge 1 any events E1015840

isin

F119881

1198611015840 E10158401015840

isin F119881

11986110158401015840 and some 119862 gt 0

10038161003816100381610038161003816P E

1015840capE

10158401015840 minus P E

1015840P E

1015840101584010038161003816100381610038161003816le eminus119862 ln2119871

(21)

One can easily check that for any 119901 gt 0 120572 isin (1 2) some120599 isin (0 1) 119886 gt 0 and 119871

0isin N large enough

eminus119862 ln2(1198710)120572119896

lt ((1198710)120572119896

)

minus119886119901(1+120579)119896

119896 ge 0 (22)

The rate of decay of correlations indicated in the RHS of(22) is required to prove dynamical localization bounds withdecay rate of EF correlators faster than polynomial and it canbe relaxed to a power-law decay if one aims to prove only apower-law decay of EF correlators

Assumptions (W1)-(W2) are sufficient for the proof ofspectral and strong dynamical localization in multiparticlesystems In particular the role of (W1) is to guaranteethe eigenvalue concentration (EVC) estimates used in theMPMSA scheme However it was discovered in [4 6] thatthe traditional EVC estimates do not provide all necessaryinformation for efficient decay bounds on the eigenfunctionsof multiparticle operators More precisely conventional EVCbounds seem so far insufficient for the proof of the exponen-tial decay of eigenfunctions and EF correlators with respect


to a norm in the configuration space of 119873-particle systemsstarting from 119873 = 3 For this reason we proposed earlier[20] a new method for comparing spectra of two stronglycorrelated multiparticle subsystems and improved the EVCestimate from [4] For the newmethod to apply one needs anadditional assumption on the random potential field whichwe will describe now

First introduce the following notations Given a finitesubset 119876 sub Z we denote by 120585

119876(120596) the sample mean of the

random field 119881 over 119876

120585119876(120596) = |119876|

minus1sum

119909isin119876

119881 (119909 120596) (23)

and define the ldquofluctuationsrdquo of119881 relative to the samplemean120578119909= 119881(119909 120596)minus120585

119876(120596)119909 isin 119876 Denote byF

119876the sigma-algebra

generated by 120578119909 119881

119910 119909 isin 119876 119910 notin 119876

We will assume that the random field 119881 fulfills thefollowing condition

(W3) There exist1198621015840 119860

1015840 119887

1015840isin (0 +infin) such that for any finite

subset 119876 subZ of cardinality |119876| and anyF119876-measur-

able function 120582 and all 119904 isin (0 1]

P 120585119876(120596) isin [120582 (120596) 120582 (120596) + 119904] le 119862

1015840|119876|

1198601015840

1199041198871015840

(24)

In the particular case of a Gaussian IID field for examplewith zero mean and unit variance it is well-known from thestandard courses of probability that 120585

119876is independent of the

ldquofluctuationsrdquo 120578119909 so it is Gaussian with variance |119876|minus1 (if

119881(119909 120596) has unit variance) that its probability density 119901119876is

bounded although 119901119876infinsim |119876|

12rarr infin as |119876| rarr infin

Property (W3) has been recently proven for a larger class ofIID potentials (cf [21])

Specifically (W3) has been established in [21] for IID ran-dom potentials with marginal probability measure 120583 whichadmits a representation of the form 120583 = 120583

1lowast 120583

2 where 120583

2is

an arbitrary probability measure onR and 1205831is supported by

some bounded interval [119886 119887] and admits probability density1205881(sdot) which fulfills the following condition

0 lt 120588lowastle 120588

1 (119905) le 120588 lt infin

1205881015840

1(119905)

1205881(119905)le 119862

1lt infin

forall119905 isin (119886 119887)

(25)

A prototypical example is the uniform distribution on [119886 119887]

25 Assumptions on the Interaction Potential We assume thatthe interaction potential 119880 generating the interaction energyU satisfies one of the following decay conditions

(U0) There exists 1199030lt infin such that

119880 (119903) = 0 forall119903 ge 1199030 (26)

(U1) There are some 119862 isin (0 +infin) 120575 isin (0 114) and 120577 isin(0 120575(1 + 120575)) such that

|119880 (119903)| le 119862eminus1198881199031minus120577

forall119903 ge 0 (27)

Naturally (U0) rArr (U1) We will prove the multiparticlelocalization first under the strongest assumption (U0) (lead-ing to a simpler proof) to illustrate the general structure ofthe DSA procedure and then extend the proof to the infinite-range interactions satisfying (U1)

26 Main Results

Theorem 2 Assume that a bounded random field 119881 fulfillsconditions (W1)ndash(W3) and the interactionU fulfills one of theconditions (U0) and (U1) There exists 119892

0isin (0 +infin) such that

for |119892| ge 1198920the following holds true

(A) With probability one the fermionic operator H(120596) =minusΔ + 119892V(120596) + U has pure point spectrum and allits eigenfunctions Ψ

119895(120596) are exponentially decaying at

infinity for each Ψ119895 some x

119895 and all x with 120588(x

119895 x)

large enough

10038161003816100381610038161003816Ψ

119895(x 120596)10038161003816100381610038161003816 le eminus119898120588(x119895x) 119898 gt 0 (28)

(B) For some 119862 119886 119888 gt 0 and any x y isinZNgt

E[sup119905isinR

10038161003816100381610038161003816⟨1y10038161003816100381610038161003816eminusi119905H(120596)10038161003816100381610038161003816

1x⟩10038161003816100381610038161003816] le 119862eminus119886 ln

1+119888120588(xy) (29)

Consequently for any finite subset K subZNgt

E[sup119905isinR

100381710038171003817100381710038171003817e(1198862) ln

1+119888Xeminusi119905H(120596)1K100381710038171003817100381710038171003817] lt infin (30)

where XK is the operator of multiplication by thefunction x 997891rarr (120588(K x) + 1)

Remark 3 The assumption of boundedness of the randompotential is inessential for the proof of spectral localization(exponential decay of all eigenfunctions) and for the analogof the decay bound (29) on eigenfunction correlators inarbitrary large but finite subset Λ sub ZN

gtof the 119873-particle

configuration space (cf Corollary 27)Without any conditionon the rate of decay of the tail probabilities P|119881(119909 120596)| ge 119904as 119904 rarr +infin the propagator eminusi119905H(120596) in (29)-(30) has to bereplaced with 119875

119868(H(120596))eminusi119905H(120596) where 119868 sub R is an arbitrary

bounded interval and 119875119868(H(120596)) is the spectral projection for

H(120596) on 119868

Remark 4 Exponential spectral localization can be provedwithout condition (W3) with (W1) relaxed to Holder conti-nuity Indeed one can use the EVCbound from [4 Lemma 2]valid for the pairs of balls distant in the so-called Hausdorffpseudo-metric dH in the multiparticle configuration space


defined as follows for the configurations x = 1199091 119909

119873

y = 1199101 119910

119873

dH (x y) = max [max119894

dZ (119909119894 y) max119894

dZ (119910119894 x)] (31)

where

dZ (119909 y) = min119895

dZ (119909 119910119895) (32)

In this case the decay bounds on the eigenfunction correla-tors would also be established with respect to dH replacingthe max-distance 120588

E[sup119905isinR

10038161003816100381610038161003816⟨1y10038161003816100381610038161003816eminusi119905H(120596)10038161003816100381610038161003816

1x⟩10038161003816100381610038161003816] le 119862 (x) eminus119886 ln

1+119888dH(xy) (33)

Observe that the RHS bound is not uniform as in (29) butdepends upon x this is an artefact of the EVC bounds basedon the Hausdorff distance (cf [4 6 7 17]) Of course theroles of x and y in the LHS of (33) are similar so the factor119862(x) in the RHS can be replaced by 119862(y) or more preciselyby some function 1198621015840

(diam (x) and diam (y)) As to the proofsthey require only minor notational modifications for theyadapt easily to any kind of pseudometric in the 119873-particleconfiguration space for which an EVC bound of the generalform (64) can be proved with some ℎ

119871(119904) le 119862119871

119860119904119899 119862119860 isin

(0 +infin) and 119887 isin (0 1]

3 Deterministic Bounds

31 Geometric Resolvent Inequality The most essential partof the Direct Scaling Analysis concerns the finite-volumeapproximations H

Λ(120596) of the random Hamiltonian H(120596)

acting in finite-dimensional spacesHΛ withΛ subZN

gt |Λ| equiv

card Λ lt infin We define

HΛ (120596) = 1

ΛH (120596) 1Λ HΛ (34)

where the indicator function 1Λis identifiedwith the operator

of multiplication by 1ΛHΛis a bounded finite-dimensional

Hermitian operator so its spectrum Σ(HΛ) is real

Operator (minusΔ) can be represented as follows

minusΔ = minus119899ZN + sum

⟨xy⟩(Γxy + Γyx)

(Γxy119891) (x) = 120575xy119891 (y) (35)

and 120575xy is the Kronecker symbolRecall that we denote byG

Λ(119864) = (H

Λminus119864)

minus1 the resolventof HΛand by G(x y 119864) the matrix elements thereof in the

standard delta-basis (the Green functions)The so-called Geometric Resolvent Inequality (GRI) for

the Green functions can be easily deduced from the secondresolvent identity (we omit 119864 for brevity)10038161003816100381610038161003816GB119871 (x y)

10038161003816100381610038161003816

le 119862ℓ

maxkisin120597minusBℓ(x)

10038161003816100381610038161003816GBℓ(x) (x k)

10038161003816100381610038161003816max

k1015840isin120597+Bℓ(x)

10038161003816100381610038161003816GB119871 (k

1015840 y)10038161003816100381610038161003816

(36)

where

119862ℓ=1003816100381610038161003816120597Bℓ

(k)1003816100381610038161003816 le 119862 (119873) ℓ119873 (37)

See for example [22] Clearly (36) implies the inequality

10038161003816100381610038161003816G

119861119871(x y)10038161003816100381610038161003816

le (119862ℓmax

120588(xk)=ℓ

10038161003816100381610038161003816GBℓ(x) (x k)

10038161003816100381610038161003816) max

120588(uk1015840)leℓ+1

10038161003816100381610038161003816GB119871 (k

1015840 y)10038161003816100381610038161003816

(38)

Sometimes we use another inequality stemming from (36)

10038161003816100381610038161003816GB119871 (x y 119864)

10038161003816100381610038161003816

le 119862ℓ

10038171003817100381710038171003817GBℓ(u) (119864)

10038171003817100381710038171003817max

k120588(uk)leℓ+1

10038161003816100381610038161003816GB119871 (k y 119864)

10038161003816100381610038161003816

(39)

Similarly for the solutions 120595 of the eigenfunction equationH120595 = 119864120595 one has with x isin B

ℓ(u)

1003816100381610038161003816120595 (x)1003816100381610038161003816 le 119862ℓ

10038171003817100381710038171003817GBℓ(x) (119864)

10038171003817100381710038171003817max

y120588(xy)leℓ+11003816100381610038161003816120595 (y)

1003816100381610038161003816 (40)

provided that 119864 is not an eigenvalue of the operatorHBℓ(u)

32 Dominated Decay Bounds for the Green Functions andEigenfunctions The analytic statements of this subsectionapply indifferently to any discrete Schrodinger operators onfairly general graphs regardless of their single- or multiparti-cle structure of the potential energy To avoid any confusionin this subsection we denote the underlying graph by G inapplications to 119873-particle models in Z one has to set G =ZN

gt

Definition 5 Fix an integer ℓ ge 0 and a number 119902 isin (0 1)Consider a finite connected subgraph Λ sub G and a ball119861119877(119906) ⊊ Λ A function 119891 119861

119877(119906) rarr R

+is called (ℓ 119902)-

dominated in 119861119877(119906) if for any ball 119861

ℓ(119909) sub Λ

119877(119906) one has

119891 (119909) le 119902 max119910d(119909119910)leℓ

119891 (119910) (41)

Lemma 6 Let be given integers 119871 ge ℓ ge 0 and a number119902 isin (0 1) If 119891 Λ rarr R

+on a finite connected graph Λ is

(ℓ 119902)-dominated in a ball 119861119871(119909) ⊊ Λ then

119891 (119909) le 119902lfloor(119871+1)(ℓ+1)rfloor

M (119891 119861)

le 119902(119871minusℓ)(ℓ+1)

M (119891 119861)

M (119891 Λ) = max119909isinΛ

1003816100381610038161003816119891 (119909)1003816100381610038161003816

(42)

Proof The claim follows from (41) by induction (ldquoradialdescentrdquo)

119891 (119909) le 119902M (119891 119861ℓ+1) le sdot sdot sdot le 119902119895M (119891 119861

119895(ℓ+1)) le sdot sdot sdot

le 119902lfloor(119871+1)(ℓ+1)rfloor

M (119891 119861)

(43)


Lemma 7 Let Λ be a finite connected graph and 119891 ΛtimesΛ rarrR

+ (1199091015840 11990910158401015840) 997891rarr 119891(1199091015840 11990910158401015840) a function which is separately (ℓ 119902)-

dominated in 1199091015840 isin 1198611199031015840(119906

1015840) ⊊ Λ and in 11990910158401015840 isin 119861

11990310158401015840(119906

10158401015840) ⊊ Λ with

d(1199061015840 11990610158401015840) ge 1199031015840 + 11990310158401015840 + 2 Then

119891 (1199061015840 119906

10158401015840) le 119902

lfloor(1199031015840+1)(ℓ+1)rfloor+lfloor(119903

10158401015840+1)(ℓ+1)rfloor

M (119891 119861)

le 119902(1199031015840+11990310158401015840minus2ℓ)(ℓ+1)

M (119891 119861)

(44)

Proof Fix any point 11991010158401015840 isin 11986111990310158401015840+1(119906

10158401015840) and consider the func-

tion

11989111991010158401015840 119910

1015840997891997888rarr 119891 (119910

1015840 119910

10158401015840) (45)

which is (ℓ 119902)-dominated in 1198611199031015840(119906

1015840) ⊊ Λ so by Lemma 6

11989111991010158401015840 (119906

1015840) = 119891 (119906

1015840 119910

10158401015840) le 119902

lfloor(1199031015840+1)(ℓ+1)rfloor

M (119891 Λ times Λ) (46)

Now introduce the function 1199061015840 119910

10158401015840997891rarr 119891(119906

1015840 119910

10158401015840) which is

(ℓ 119902)-dominated in 11986111990310158401015840(119906

10158401015840) ⊊ Λ and bounded by the RHS of

(46) Applying again Lemma 6 the claim follows

The relevance of the above notions and results is illus-trated by Lemma 10 stated in the next subsection It couldbe stated in a general form for arbitrary countable graphsbut its formulation relies on the notions of ldquononsingularrdquoand ldquononresonantrdquo balls For our model these are given inDefinitions 8 and 9 However the proof given in [14] (cf [14Lemma 4]) applies indifferently to arbitrary graphs

33 Localization and Tunneling in Finite Balls From thispoint on we will work with a sequence of length scalespositive integers 119871

119896 119896 ge 0 defined recursively by 119871

119896+1=

lceil119871120572

119896rceil 119871

0gt 2 For clarity we keep the value 120572 = 43 observe

that 1205722 lt 2 In several arguments we will require the initialscale 119871

0to be large enough

Definition 8 Given a sample of the random potential119881(sdot 120596)a ball B

119871(u) is called

(i) 119864-nonresonant (119864-NR) if GB(119864 120596) le e+119871120573

and 119864-resonant (119864-R) otherwise

(ii) completely 119864-nonresonant (119864-CNR) if it does notcontain any 119864-R ball B

ℓ(u) sube B

119871(u) with ℓ ge 1198711120572

and 119864-partially resonant (119864-PR) otherwise

Definition 9 Given a sample 119881(sdot 120596) a ball B119871(u) is called

(119864119898)-nonsingular ((119864119898)-NS) if

maxyisin120597minusB119871(u)

10038161003816100381610038161003816GB119871(u) (u y 119864 120596)

10038161003816100381610038161003816le eminus120574(119898119871)119871+2119871

120573

(47)

where

120574 (119898 119871) = 119898 (1 + 119871minus120591) 120591 =

1

8 (48)

Otherwise it is called (119864119898)-singular ((119864119898)-S)

Lemma 10 (cf [14 Lemma 4]) Consider a ballB119871119896(u) and an

operatorH = HB119871119896 (u)with fixed (nonrandom) potentialW Let

120595119895 119895 = 1 |B

119871119896(u)| be the normalized eigenfunctions of

H Pick a pair of points x0 y

0isin B

119871119896(u)withd(x

0 y

0) gt 119871

119896minus1+1

and an integer

119877 isin [119871119896minus1 d (x

0 y

0) minus 119871

119896minus1+ 1] (49)

Suppose that any ball B119871119896minus1(k) sub B

119877(x

0) is (119864119898)-NS and set

119902 = eminus120574(119898ℓ)ℓ (50)

Then

(A) the kernel Π120595119895(x y

0) of the spectral projection Π

120595119895=

|120595119895⟩⟨120595

119895| considered as a function of x is (ℓ + 1 119902)-

dominated in x isin B119877(x

0) and globally bounded by 1

(B) if the ball B119871119896(u) is also 119864-NR then the Green function

B119877(x

0) ni x 997891997888rarr

100381610038161003816100381610038161003816GB119871119896 (u)

(x y 119864)100381610038161003816100381610038161003816

(51)

is (ℓ + 1 119902)-dominated in x isin B119877(x

0) and globally

bounded by e119871120573

By Hermitian symmetry of the Green functions thecounterparts of assertions (A) and (B) relative to the secondargument y isin B

119877(y

0) also hold true with x = x

0fixed

In the next definition we use a parameter 984858 = (120572 minus 1)2with 120572 = 43 one obtains 984858 = 16 and (1 + 984858)120572 = 78

Definition 11 A ball B119871(u) is called 119898-localized (119898-Loc

in short) if every eigenfunction 120595119895of the operator HB119871(u)

satisfies10038161003816100381610038161003816120595

119895(x)120595

119895(y)10038161003816100381610038161003816 le eminus120574(119898119871)120588(xy) (52)

for any points x y isin B119871(u) with 120588(x y) ge 119871(1+984858)120572 equiv 11987178

Definition 12 A pair of balls B119871(x) B

119871(y) is called distant iff

120588(x y) ge 119862119873119871 with 119862

119873= 2119860

119873+ 3 119860

119873= 4119873

(The role of the constants 119860119873and 119862

119873will become clear

in Section 43)

Definition 13 A ball Bℓ120572(u) is called 119898-tunneling (119898-T) if it

contains a pair of distant119898-NLoc balls Bℓ(x) B

ℓ(y) and119898-

nontunneling (119898-NT) otherwise

It is to be emphasized that unlike the property of 119864-resonance or (119864119898)-singularity the tunneling property is notrelated to a specific value of energy 119864 and tunneling even in asingle ball occurswith a small probabilityThis is an advantageof the DSA compared to the fixed-energy MSA yet as wassaid in Introduction the DSA provides final decay boundsfor the Green functions in an entire range of energies likethe variable-energyMSAHowever unlike theMSA theDSAalso provides bounds on the eigenbases in the finite balls

Lemma 14 Let be given 119898-localized ball B = B119871(u) If for

some 119864 isin R B is also 119864-NR then it is (119864119898)-NS


Proof The matrix elements of the resolvent GB(119864) can beassessed as follows

1003816100381610038161003816GB (x y 119864)1003816100381610038161003816 le sum

119864119895isinΣ(HB)

10038161003816100381610038161003816120595

119895(x)1003816100381610038161003816100381610038161003816100381610038161003816120595

119895(y)10038161003816100381610038161003816

10038161003816100381610038161003816119864 minus 119864

119895

10038161003816100381610038161003816

(53)

Here Σ(HB) is the spectrum of the operatorHBIf dist (119864 Σ(HB)) ge eminus119871

120573

and ln (2119871 + 1)119873 le 119871120573 then the119898-Loc property implies

1003816100381610038161003816GB (x y 119864)1003816100381610038161003816 le eminus120574(119898119871)119871+119871

120573+ln|B|

le eminus120574(119898119871)119871+2119871120573

(54)

Observe that with 119898 ge 1 and 1 minus 120591 gt 120573 one has for 1198710

large enough (hence 119871119896large enough 119896 ge 0)

119898(1 + 119871minus120591

119896) 119871

119896minus 2119871

120573

119896= 119898119871

119896+ 119898119871

1minus120591

119896minus 2119871

120573

119896

ge 119898(1 +1

2119871minus120591

119896) 119871

119896

(55)

We will see that the condition 119898 ge 1 is fulfilled for |119892| largeenough (cf Lemma 17)

Notice that the above proof uses the following fact thecardinality of an 119873-particle ball of radius 119871 is polynomiallybounded in L This will be used in Section 8 The sameremains true for the symmetric powers of graphsZ satisfyingthe growth condition (129) this includes all latticesZ = Z119889119889 ge 1

Lemma 15 There is (1) isin N such that for 1198710ge

(1)

(A) if a ball B119871119896+1(u) is 119864-CNR and contains no pair of

distant (119864119898)-S balls B119871119896(k) B

119871119896(w) then it is also

(119864119898)-NS(B) if a ball B

119871119896+1(u) is 119898-NT and 119864-CNR then it is also

(119864119898)-NS

Proof (A) By assumption either B119871119896+1(u) is 119864-CNR and

contains no (119864119898)-S ball of radius 119871119896or there is a point

w isin B119871119896+1(u) such that any ball B

119871119896(k) sub B

119871119896(u) with

120588(w k) ge 119862119873119871119896is (119864119898)-NS We will obtain the required

bound without using the balls with 120588(w k) lt 119862119873119871119896

In the former case such an exclusion is unnecessary butin order to treat both situations with one argument we canformally set w = u (in fact the choice of the center of aldquofictitiousrdquo singular ball does not really matter)

Fix arbitrary points x y with 119877 = 120588(x y) gt 1198711+984858119896

asbefore 984858 = 16 By the triangle inequality

120588 (xB(119862119873minus1)119871119896

(w)) + 120588 (yB(119862119873minus1)119871119896(w))

ge 119877 minus (2119862119873minus 2) 119871

119896

(56)

Assume first that

1199031015840= 120588 (xB

(119862119873minus1)119871119896(w)) ge 119871

119896+ 1

11990310158401015840= 120588 (yB

(119862119873minus1)119871119896(w)) ge 119871

119896+ 1

(57)

Inside B1199031015840(x) cup B

11990310158401015840(y) all the balls of radius 119871

119896are auto-

matically disjoint from B2119871119896(w) thus they are (119864119898)-NS

Furthermore

1199031015840+ 119903

10158401015840ge 119877 minus 2 (119862

119873minus 1) 119871

119896minus 2 ge 119877 minus 2119862

119873119871119896 (58)

Consider the set B = B1199031015840(x) times B

11990310158401015840(y) and the function

119891 B rarr Cdefined by119891(x1015840 x10158401015840) = GB119871119896+1(x1015840 x10158401015840 119864) Since119864 is

not a pole of the resolventGB119871119896+1 (u)(sdot) its matrix elements are

well defined hence bounded on a finite set By Lemma 10119891 is(119871

119896 119902)-dominated both in x1015840 and in x10158401015840 with 119902 le eminus120574(119898119871119896 119899)

Therefore one can write with the convention minusln 0 = +infinusing Lemma 7

minus ln119891 (u y)

ge minusln [(eminus119898(1+(12)119871minus120591119896 )119871119896)

(119877minus2119862119873119871119896minus2119871119896)(119871119896+1)

e119871120573

119896+1]

= 119898(1 +1

2119871minus120591

119896)119871119896

119871119896+ 1119877 (1 minus 3C

119873119877minus1119871119896) minus 119871

120573

119896+1

= 119898119877[(1 +1

2119871minus120591

119896) (1 minus 119871

minus1

119896) (1 minus 3119862

119873119871minus984858

119896)

minus119871120573

119896+1

119898119877] ge 119898(1 +

1

4119871minus120591

119896)119877 ge 120574 (119898 119871

119896+1) 119877

(59)

If 1199031015840 = 0 (resp 11990310158401015840 = 0) the required bound follows from thedominated decay of the function 119891(x1015840 x10158401015840) in x10158401015840 (resp in x1015840)

(B)Assume otherwiseThen by assertion (A) the119864-CNRball B

119871119896+1(u) must contain a pair of distant (119864119898)-S balls

B119871119896(x) B

119871119896(y) Both of them are 119864-NR since B

119871119896+1(u) is 119864-

CNR By virtue of Lemma 14 both B119871119896(x) and B

119871119896(y) must

be119898-NLoc so thatB119871119896+1(u)must be119898-T which contradicts

the hypothesis

Lemma 16 There is (2) isin N such that for all 1198710ge

(2) if forany 119864 isin R a ball B

119871119896+1(u) contains no pair of distant (119864119898)-S

balls of radius 119871119896 then it is119898-Loc

Proof One canproceed as in the proof of the previous lemmabut with the functions 119891

119895 (k1015840 k10158401015840) 997891rarr |120595

119895(k1015840)120595

119895(k10158401015840)| where

120595119895 119895 isin [1 |B

119871119896+1(u)|] are normalized eigenfunctions of oper-

atorHB119871119896+1 (u) Notice that the 119864

119895-nonresonance condition for

B119871119896+1(u) is not required here since one has 120595

119895 = 1 so the

function 119891119895is globally bounded by 1 Let x y isin B

119871119896+1(u) and

assume 120588(x y) = 119877 ge 1198711+984858119896

Arguing as in the previous lemma and using the domi-

nated decay of the function 119891119895(sdot sdot) (cf Lemma 10) we obtain

for any pair of points with 120588(y1015840 y10158401015840) ge 1198711+984858119896

minus ln 10038161003816100381610038161003816120595119895 (x)120595119895 (y)10038161003816100381610038161003816


(119877minus2119862119873119871119896minus2119871119896)(119871119896+1)

]

(60)

A direct comparisonwith the RHS of the first equation in (59)shows that the RHS of (60) is bigger owing to the absence of


the large factor e119871120573

119896 Therefore it admits the same (or better)lower bound as in (59)

Below we summarize the relations between the parame-ters used in the above proof

120588 (x y) ge 1198711+984858119896minus1

0 lt 120591 lt 120588

1 + 984858 lt 120572

120573 lt 1 minus 120591

(61)

These conditions are fulfilled for example with

120572 =4

3

120588 =1

6

120591 =1

8

120573 =1

2

(62)

4 Scaling Analysis of EigenfunctionsFinite-Range Interaction

41 Initial Scale Estimates

Lemma 17 (initial scale bound) For any 1198710gt 2 119898 ge 1 and

119901 gt 0 there is 1198920= 119892

0(119898 119901 119871

0) lt infin such that for all 119892 with

|119892| ge 1198920 any ball B

1198710(119906) and any 119864 isin R

P B1198710(u) 119894119904 (119864119898) -119878 le 119871minus119901

0

P B1198710(u) 119894119904 119898-NL119900119888 le 119871

minus119901

0

(63)

See the proof in the Appendix

42 EVC Bounds for Distant Pairs of Balls The followingstatement reflects the progress achieved in the area of mul-tiparticle eigenvalue concentration (EVC) estimates since thetime when manuscript [18] had appeared

Theorem 18 (cf [21 Theorem 4] [19 Lemma 24]) Let 119881 ZtimesΩ rarr R be a randomfield satisfying (W3) with parameters119887 119887

1015840isin (0 1] 1198601015840

isin (0 +infin) For all 119871 gt 0 large enough anydistant pair of balls B(119873)

119871(u1015840) B(119873)

119871(u10158401015840) the following bound

holds for any 119904 isin (0 1]

P dist (Σ119871u1015840 Σ119871u10158401015840) le 119904 le (3119871)

2119873ℎ119871 (2119904) (64)

where Σ119871u1015840 Σ119871u10158401015840 are the respective spectra of HB119871(u1015840) and

HB119871(u10158401015840) and

ℎ119871 (119904) = 119862

1015840101584011987111986010158401015840

119904119887and1198871015840

119887 and 1198871015840equiv min [119887 1198871015840] (65)

and 11986010158401015840= 119860

10158401015840(119860

1015840 119873) isin (0 +infin) Consequently this implies

the following bound for 1205731015840 isin (0 120573) all 119871 large enough and anypair u1015840 u10158401015840 with d(u1015840

u10158401015840) gt 4119873119871

P exist119864 isin R B119871(u1015840) B

119871(u10158401015840) 119886119903119890 119864-119875119877 le eminus119871

1205731015840

(66)

A considerable advantage of the above EVC boundcompared to [4 Lemma 2] is that it holds for any distant pairof balls In [4] as well as in [17] a similar bound is establishedonly for pairs of balls sufficiently distant with respect to theso-called Hausdorff (pseudo)metric such a bound did notcover some pairs distant with respect to the more naturalsymmetrized distance like 120588 As a result one could not proveuniform decay of eigenfunction correlators in the physicallyrelevant distance in the multiparticle configuration space Asimilar difficulty was encountered in [6]

43 Partially Interactive119873-Particle Balls

Definition 19 An 119873-particle ball B119871(u) is called partially

interactive (PI in short) if diam u gt 119860119873119871 119860

119873= 4119873 and

fully interactive (FI) otherwise

The local HamiltonianH(119873)

B119871(u)with a finite-range interac-

tion in a PI ball is algebraically decomposable as follows

H(119873)

B119871(u) = H(1198991015840)

B119871(u1015840)otimes 1(119899

10158401015840)+ 1(119899

1015840)otimesH(119899

10158401015840)

B119871(u10158401015840)(67)

with 1198991015840+11989910158401015840 = 119873 u = (u1015840 u10158401015840) u1015840isinZn1015840

gt u10158401015840isinZn10158401015840

gt while for

a fully interactive ball one cannot guarantee such a property(In the case of a rapidly decaying interaction the RHS is closein norm to the LHS)

Lemma 20 (A) If B119871(x) is PI then there exists a partition of

[[1119873]] into nonempty complementary subsetsJJ119888 such that120588(ΠJB119871

(x) ΠJ119888B119871(x)) gt 2119871

(B) If B119871(x) B

119871(y) are two FI balls with 120588(x y) gt 11119873119871

then 120588(ΠB119871(x) ΠB

119871(y)) ge 2119871

Here the value 11119873 is obtained as 2119860119873+ 3

See the proof in the Appendix Section A2Now we state a result on localization in PI balls for

the Hamiltonians with a short-range interaction (condition(U0))The reader may want to compare the elementary proofof Lemma 21 below with that of Lemma 3 in [4] which issubstantially more complex The simplification comes herefrom the fact that we rely on the 119898-Loc property that isexponential decay of all EFs of a Hamiltonian in a given balland this property is nonrelated to a specific energy but coversthe entire spectrum of the Hamiltonian at hand this resultsin a few-line proof of Lemma 21 as well as of an importantprobabilistic bound (Lemma 22) stemming from it

Lemma21 Assume that the interactionU has finite range 1199030isin

[0 +infin) (cf (U0)) Consider a PI119873-particle ball with canonicalfactorization

B(119873)

119871119896(u) = B(119899

1015840)

119871119896(u1015840) times B(119899

10158401015840)

119871119896(u10158401015840) (68)


If the balls B(1198991015840)

119871119896minus1(u1015840) and B(119899

10158401015840)

119871119896minus1(u10158401015840) are 119898-Loc then the ball

B(119873)

119871119896(u) is also119898-Loc

Proof The interaction between subsystems corresponding tou1015840 and u10158401015840 vanishes soHB119871119896 (u)

admits the decomposition

HB119871119896 (u)= HB(119899

1015840)119871119896

(u1015840) otimes 1(11989910158401015840)+ 1(119899

1015840)otimesHB(119899

10158401015840)119871119896

(u10158401015840) (69)

Thus the eigenfunctions Ψ119895of HB119871119896 (u)

can be chosen in theformΨ

119895= 120601

1198951015840 otimes120595

11989510158401015840 where 120601

1198951015840 are eigenfunctions ofHB(119899

1015840)119871119896

(u1015840)and respectively 120595

11989510158401015840 are eigenfunctions of HB(119899

10158401015840)119871119896

(u10158401015840) Now

the required decay bounds on the projection kernel10038161003816100381610038161003816Ψ

119895 (x)Ψ119895 (y)10038161003816100381610038161003816

=100381610038161003816100381610038161206011198951015840 (x1015840)120601

11989510158401015840 (x10158401015840)120595

1198951015840 (y1015840)120595

11989510158401015840 (y10158401015840)10038161003816100381610038161003816

(70)

follow directly from the respective bounds on 120601 and120595 whichare guaranteed by the 119898-Loc hypotheses on the projectionballs B(119899

1015840)

119871119896(u1015840) and B(119899

10158401015840)

119871119896(u10158401015840)

Given numbers 119901 120579 gt 0 introduce the quantities

119875 (119899 119896) = 2119873minus119899119901 (1 + 120579)

119896 1 le 119899 le 119873 119896 ge 0 (71)

Later it will be required that 120599 isin (0 radic2 minus 1)

Lemma 22 Suppose that for all 119899 isin [1119873 minus 1] and for a given119896 ge 0 the following bound holds for any 119899-particle ball B(119899)

119871119896(u)

P B(119899)

119871119896(u) 119894119904 119898-NL119900119888 le 119871

minus119875(119899119896)

119896 (72)

Then for any119873-particle PI ball B(119873)

119871119896(u) one has

P B(119873)

119871119896(u) 119894119904 119898-NL119900119888 le 2119871

minus2119875(119873119896)

119896le1

4119871minus119875(119873119896)

119896 (73)

Proof It follows from Lemma 21 that a ballB(119873)

119871119896(u) admitting

a factorization of the form (69) is 119898-NLoc only if at leastone of the projection balls B(119899

1015840)

119871119896(u1015840) B(119899

10158401015840)

119871119896(u10158401015840) is 119898-NLoc

Therefore with 1198710large enough we obtain

P B(119873)

119871119896(u) is 119898-NL119900119888

le P B(11989910158401015840)

119871119896(u10158401015840) is 119898-NL119900119888

+ P B(11989910158401015840)

119871119896(u10158401015840) is 119898-NL119900119888

le 119871minus119875(1198991015840119896)

119896+ 119871

minus119875(11989910158401015840119896)

119896le 2119871

minus2119875(119873119896)

119896lt1

4119871minus119875(119873119896)

119896

(74)

44 Fully Interactive 119873-Particle Balls Owing to Lemma 22it remains to establish localization bounds for 119873-particle FIballs B

119871119896+1(x) assuming if necessary similar bounds

(i) For 119899-particle balls of any radius 1198711198961015840 1198961015840 ge 0 with 119899 lt

119873(ii) For119873-particle balls of any radius 119871

1198961015840 with 1198961015840 le 119896

Lemma 23 (main inductive lemma) Suppose that for a given119896 ge 0 and all 119894 isin [0 119896] the following bound holds for any 119873-particle ball B(119873)

119871119896(u)

P B(119873)

119871 119894(u) 119894119904 119898-NL119900119888 le 119871

minus119875(119873119894)

119894 (75)

Then for any119873-particle FI ball B(119899)

119871119896+1(x) one has

P B(119873)

119871119896+1(u) 119894119904 119898-NL119900119888 le 119871

minus119875(119873119896+1)

119896+1 (76)

provided that

119901 gt2120572

2

2 minus 1205722119873119889

0 lt 3120579 le min2 minus 1205722

1205722minus2119873119889

119901radic2 minus 1

(77)

Furthermore for any pair of distant balls B(119873)

119871119896(u) B(119873)

119871119896(k) one

has for some 119886 119888 gt 0

P exist119864 isin R B(119873)

119871119896(u) B(119873)

119871119896(k) 119886119903119890 (119864119898) -119878

le 119871minus119875(119873119896)

119896equiv 119871

minus119901(1+120599)119896

119896le eminus119886 ln

1+119888119871119896

(78)

Proof Introduce the following events

N119896+1= B(119873)

119871119896+1(u) is 119898-NLo119888

S(2)

119896

= exist119864exist distant (119864119898) -S balls B(119873)

119871119896(u) B(119873)

119871119896(k)

sub B(119873)

119871119896+1(w)

R(2)

119896= exist119864exist distant 119864-PR balls B(119873)

119871119896(u) B(119873)

119871119896(k)

sub B(119873)

119871119896+1(w)

(79)

By virtue of Lemma 16 we have

N119896+1sub S

(2)

119896subR

(2)

119896cup (S

(2)

119896R

(2)

119896) (80)

and by Theorem 18 PR(2)

119896 le eminus119871

1205731015840

119896 so it remains to assessthe probability PS(2)

119896R

(2)

119896 Fix points u k isin B(119873)

119871119896+1(w) and

introduce the event (figuring in (78))

S(2)

119896(u k) = exist119864 B(119873)

119871119896(u) B(119873)

119871119896(k) are (119864119898) -S (81)

Within the event S(2)

119896(u k) R(2)

119896 either B(119873)

119871119896(u) or B(119873)

119871119896(k)

must be 119864-CNR wlog assume that B(119873)

119871119896(u) is 119864-CNR

Since it is (119864119898)-S Lemma 15 implies that it must contain a


pair of 119898-NL119900119888 balls B(119873)

119871119896minus1(y1015840) B(119873)

119871119896minus1(y10158401015840) with d(y1015840 y10158401015840) gt

(12)1198711+120575

119896minus1 Consider two possible situations

(1) Both B(119873)

119871119896minus1(y1015840) and B(119873)

119871119896minus1(y10158401015840) are FI

Apply Lemma 20 assertion (B)

120588 (ΠB119871119896minus1(u) ΠB

119871119896minus1(k)) ge 2119871

119896 (82)

Using the inductive assumption (75) and the mixing condi-tion (W2) we have with 119871

119896minus1= 119871

11205722

119896+1and 119871

0large enough

P B(119873)

119871119896minus1(y1015840) B(119873)

119871119896minus1(y10158401015840) are 119898-NL119900119888

le 2119871minus(2119901120572

2)(1+120579)

119896minus1

119896+1

(83)

The number of such pairs y1015840 y10158401015840 is bounded by(12)|B(119873)

119871119896(w)|2 le (321198732)1198712119873

119896 so in this case

P S(2)

119896R

(2)

119896 le32119873

2119871minus(2119901120572

2)(1+120579)

119896minus1+2119873

119896+1 (84)

A straightforward calculation shows that for 1198710

largeenough the RHS is bounded by (14)119871minus119875(119873119896+1)

119896+1 under con-

ditions (77) Indeed we need the inequality

2119901

12057222119873minus119873119901 (1 + 120579)

119896minus1minus 2119873119889 gt 2

119873minus119873119901 (1 + 120579)

119896+1 (85)

or equivalently

(1 + 120579)2lt2119901

1205722minus

2119873119889

119901 (1 + 120579)119896minus1 (86)

By construction 120599 isin (0 1) so one has (1 + 120579)2 lt 1 + 3120599and 2119873119889119901 gt 2119873119889(119901(1 + 120579)119896minus1) Therefore if 119901 gt (21205722(2 minus1205722))119873 and

3120599 le2 minus 120572

2

1205722minus2119873

119901 (87)

then (85) is also satisfied so that

P S(2)

119896 le P R

(2)

119896 + P S

(2)

119896R

(2)

119896

le eminus1198711205731015840

119896 +1

4119871minus119875(119873119896+1)

119896+1lt1

2119871minus119875(119873119896+1)

119896+1

(88)

yielding assertion (76) in case (1)(2) EitherB(119873)

119871119896minus1(y1015840) orB(119873)

119871119896minus1(y10158401015840) is PIThen by Lemma 22

P B(119873)

119871119896minus1(y1015840) B(119873)

119871119896minus1(y10158401015840) is 119898-NL119900119888

le min (P B(119873)

119871119896minus1(y1015840) is 119898-NL119900119888

P B(119873)

119871119896minus1(y10158401015840) is 119898-NL119900119888) le 119871

minus2119901(1+120579)119896minus1

119896minus1

lt 119871minus119901(1+120579)

119896+1

119896+1

(89)

provided that (1 + 120579)2 lt 2 that is 120579 lt radic2 minus 1

Finally a straightforward calculation shows that for some119886 119888 gt 0 one has the inequality

119871minus119875(119873119896)

119896equiv 119871

minus119901(1+120599)119896


1+119888119871119896 (90)

This completes the proof

Applying Lemmas 17 and 22 we come by induction to thefollowing result

Theorem 24 (localization at any scale) Assume that |119892| islarge enough so that (63) hold true with119901 gt (21205722(2minus1205722))119873119889Then for any 119896 ge 0 and any u isin Z119873119889

P B(119873)

119871119896(u) 119894119904 119898-NL119900119888 le

1

4119871minus119901(1+120579)

119896

119896(91)

with 120579 gt 0 satisfying (77) In otherwords with probabilityge 1minus(14)119871

minus119901(1+120579)119896

119896 for every eigenfunction Ψ

119895of operator HB(119873)119871119896 (u)

and for all x y isin B(119873)

119871119896(u) such that x minus y ge 119871(1+120575+120588)120572

119896one

has10038161003816100381610038161003816Ψ119895(x) Ψ

119895(y)10038161003816100381610038161003816 le eminus120574(119898119871119896)119871119896 lt eminus119898119871119896 (92)

Theorem 24 marks the end of the Direct Scaling Analysisof localized eigenfunctions in arbitrarily large finite balls

5 Exponential Decay of Eigenfunctions

The DSA provides exponential decay bounds on the eigen-bases in finite balls not just on the Green functions whichshows that quantum transport in an arbitrarily large finitesystem over distances comparable with the size of the systemis strongly inhibited The proof of exponential decay of EFsin an infinitely extended configuration space requires anadditional argument originally proposed by Frohlich et al[9] and reformulated by von Dreifus and Klein [10] Morethan 20 years after [10] it is merely a simple exercise

Theorem 25 For P-ae 120596 isin Ω every normalized eigenfunc-tionΨ of operatorH(120596) satisfies the following bound for some119877(120596) x(120596) and all y with y ge 119877(120596)

1003816100381610038161003816Ψ (y)1003816100381610038161003816 le eminus119898y

(93)

Proof Pick an arbitrary vertex z isin ZNgt By the Borel-

Cantelli lemma combined with the probabilistic bound fromLemma 23 there is a subsetΩ1015840

sub ΩwithPΩ1015840 = 1 such that

for any 120596 isin Ω1015840 and some 1198960(120596) all 119896 ge 119896

0 and any 119864 isin R

there is no pair of 1198711+120575119896

-distant (119864119898)-S ballsB119871119896(x)B

119871119896(y) sub

B119871119896+2(z) Fix 120596 isin Ω1015840

Let Ψ be a normalized eigenfunction of H(120596) witheigenvalue 120582 Since Ψ

2le 1 implies Ψ

infinle 1 there is

a point x such that Ψinfin= |Ψ(x)| If

119899isin B

119871119896minus1(0) then

B119871119896(0) must be (120582119898)-S otherwise the (120582119898)-NS property

would lead to a contradiction

Ψinfin = |Ψ (x)| le eminus119898119871119896Ψinfin lt Ψinfin (94)


Thus any ball B119871119896(y) sub B

119871119896+2(z) with 120588(y z) isin [119871

119896+1 119871

119896+2)

(hence with 120588(y z) ge 119871120572119896gt 2119871

1+120575

119896) is (120582119898)-NS Note that

the function x 997891rarr |Ψ119899(x)| is (119871

119896 119902)-dominated in B

119877(y) with

119877 = 120588(y z)minus21198711+120575119896minus1 gt (12)119871

1+2120575

119896and 119902 = eminus120574(119898119871119896)119871119896 Recall

that 120591 = 1205754 (cf (62)) By Lemma 6 for 119871119896large enough

minusln 1003816100381610038161003816Ψ (y)

1003816100381610038161003816

120588 (y z)ge 119898 (1 + 119871

minus120591

119896) (1 minus

2119871119896+ 1

120588 (y z))

ge 119898(1 +1

2119871minus120591

119896) gt 119898

(95)

yielding assertion (93)

6 EF Correlators and Dynamical Localization

61 Bounds on the EFC in Finite Volumes We use herea finite-volume variant of a ldquosoftrdquo argument proposed byGerminet and Klein in [11] and adapted in our earlier papersto finite-volume Hamiltonians both single-particle (cf [14Memma 9]) and 119873-particle (cf [19 Lemma 38]) Workingwith finite volumes allows us to avoid a functional-analyticcomplement regarding the weighted Hilbert-Schmidt normsof spectral projections of operators H(120596) in the entire graphZN

gtand replace it with a simple application of Besselrsquos

inequalityDenote byB

1(119868) the set of all Borel functions 120601 R rarr C

with supp 120601 sub 119868 and 120601infinle 1

Proposition 26 (cf [14 Lemma 9] [19 Lemma 38]) Fix aninteger 119871 isin Nlowast and assume that the following bound holds for agiven pair of disjoint balls B

119871(119909) B

119871(119910) and an interval 119868 sube R

P exist 119864 isin 119868 B119871(x) B

119871(y) 119886119903119890 (119864119898) -S le 119891 (119871) (96)

for some 119891(119871) Then for any connected subgraph Λ containingthe union B

119871(119909) cup B

119871(119910) one has

E[ sup120601isinB1(119868)

10038161003816100381610038161003816⟨1x1003816100381610038161003816120601 (HΛ (120596))

1003816100381610038161003816 1y⟩10038161003816100381610038161003816]

le 119862119871119889119890minus119898119871+ 119891 (119871)

(97)

Taking into account Lemma 23 establishing the proba-bilistic bound for the quantity 119891(119871) figuring in the RHSof (96) and (97) we come to the following corollary ofProposition 26

Corollary 27 Under the hypotheses of Theorem 2 for anyx y isin ZN

gtwith d(x y) gt 2119871 + 1 any connected subset

Λ sup B119871(119909) cup B

119871(119910)

E[ sup120601isinB1(R)

10038161003816100381610038161003816⟨1x1003816100381610038161003816120601 (HΛ (120596))

1003816100381610038161003816 1y⟩10038161003816100381610038161003816] le 119862119890

minus119886 ln1+119888120588(xy) (98)

62 Dynamical Localization in an Infinite Graph Here wefollow the same path as in earlier works by Aizenman et al(cf eg [13 23])

Recall that if the random potential 119881 is as bounded thenorm of H(120596) is as bounded by a constant so its spectrumis contained in some nonrandom bounded interval 119868 Givenan interval 119868 sub R denote by C

1(119868) the set of all continuous

functions 120601 on R with supp120601 sub 119868 and 120601infinle 1

Theorem 28 Consider the Hamiltonian H(120596) of form (1)with a bounded random potential satisfying assumptions(W1)ndash(W3) and the interaction potential satisfying one of theassumptions (U0) and (U1) There is 119892

0lt +infin such that if

|119892| ge 1198920 then the following property holds true

For any bounded interval 119868 sub R containing the asspectrum of H(120596) and for all x y isin ZN

gt x = y and with the

same 119888 119886 gt 0 as in (98)

E[ sup120601isinC1(119868)

10038161003816100381610038161003816⟨1x1003816100381610038161003816120601 (H (120596))

1003816100381610038161003816 1y⟩10038161003816100381610038161003816] le 119862eminus119886 ln

1+119888120588(xy) (99)

Proof We prove the claim under assumption (U0) usingCorollary (97) established under this assumption Similarlycase (U1) will follow from the analogous results proved inSection 7 for the infinite-range interactions

Since both119881 areU are bounded the norm of the randomoperator H(120596) is as bounded by a nonrandom constanthence there exists a nonrandom bounded interval 119868 sub R

containing as the spectrum Σ(H(120596)) For any ball B and anypoints x y isin B introduce a spectral measure 120583xyB120596 uniquelydefined by

int120601 (120582) 119889120583xyB120596 (120582)

= ⟨1x1003816100381610038161003816 120601 (HB (120596))Π119868

(HB (120596))100381610038161003816100381610038161y⟩

(100)

where 120601 is an arbitrary bounded continuous function withsupp120601 sub 119868 and similar spectral measures 120583xy

120596for the operator

H(120596) on the entire graphZNgt If B

119871119896 is a growing sequence

of balls then 120583xyB119871119896 120596converge vaguely to 120583xy

120596as 119896 rarr infin Note

that this fact holds true in a much more general context ofunbounded operators by a well-known result (cf [24]) forthe strong resolvent convergence of operatorsH

119899rarr H with

a common coreD it suffices thatH119899120601 rarr H120601 for any element

120601 isin D in turn this implies the vague convergence of thespectral measures ⟨120593 120601(H

119899)120595⟩ In our case we can choose as

a core the subspace of all compactly supported functions andon such functions the operatorsH

119899converge by stabilization

So by Fatoursquos lemma on convergent measures for anymeasurable setE sub R and any growing sequence of ballsB

119871119896

E [1003816100381610038161003816120583

xy120596

1003816100381610038161003816 (E)] le lim inf119896rarrinfin

E [100381610038161003816100381610038161003816120583xyB119871119896

100381610038161003816100381610038161003816(E)] (101)

Therefore the uniform bounds in finite volumes B119871119896 estab-

lished in Corollary 27 remain valid in the entire infinitegraph

Now assertion (B) of the mainTheorem 2 for finite-rangeinteractions follows by taking continuous functions of theform

120601119905 120582 997891997888rarr 119891

119868(120582) ei119905120582 119905 isin R (102)


where 119891119868 R rarr [0 1] is an arbitrary continuous function

interpolating between 1 on an interval 119868 = [minus119877 119877] containingas the spectrum ofH(120596) and 0 on R [minus119877 minus 1 119877 + 1]

Note also that arguing as in [23 Sect 25] one canextend bound (99) to the compactly supported boundedBorel functions 120601

7 Adaptation to Infinite-Range Interactions

In this section we need to adapt a certain number ofdefinitions and statements to rapidly decaying infinite-rangeinteractions satisfying assumption (U1) In particular

(i) given a number 120575 isin (0 114) figuring in (U1) we setnow

984858 = 2120575

120572 = 1 + 4120575

120591 =120575

2

(103)

and observe that 1205722 lt 2 984858 minus 120575 gt 120591(ii) we replace the decay exponent 120574(119898 119871) = 119898(1 + 119871minus120591)

figuring in Definition 9 (cf (48)) by

120574 (119898 119871 119899) = 119898 (1 + 119871minus120591)119873minus119899+1

(104)



120588(x y) ge 1198621198731198711+120575 with 119862

119873= 4119873

Definition 30 An 119873-particle ball B119871(x) is called partially

interactive (PI) if diamu gt 21198711+120575 and fully interactive (FI)otherwise



(x) ΠJ119888B119871(x)) gt 1198711+120575 Consequently B(119873)

119871(x) admits

a factorization

B(119873)

119871(x) = B(119899

1015840)

119871(x1015840) times B(119899

10158401015840)

119871(x10158401015840) (105)

with dist(x1015840 x10158401015840) gt 21198711+120575(B) If B

119871(x) B


then 120588(ΠB119871(x) ΠB

119871(y)) ge 1198711+120575

The proof repeats verbatim that of Lemma 20 inSection A2 but with 119871 replaced by 1198711+120575

Themost important technicalmodification is required forLemma 21 the proof of whichwas very elementary due to theuse of eigenfunctions

Next introduce the following finite-range approxima-tionsU

119877119877 ge 119903

0 of a given interactionU generated by 2-body

interaction potentials 119880(2)(119903)

119880(2)

119877(119903) = 1

119903le119877119880

(2)(119903) (106)

Then for any configuration x = (x1015840 x10158401015840) isin Zn1015840gttimes Zn10158401015840

gt

with 120588(x1015840 x10158401015840) gt 119877 the truncated interaction decomposes

as follows U119877(x) = U

119877(x1015840) + U

119877(x10158401015840) As a result for any

factorized subset of the form B = B1198711015840(x1015840) times B

11987110158401015840(x10158401015840) with

120588(B1198711015840(x1015840)B

11987110158401015840(x10158401015840)) gt 119877 the truncated Hamiltonian H(119877)

Badmits the algebraic decomposition

H(119877)

B = H(119877)

B1198711015840(x1015840) otimes 1

(11989910158401015840)+ 1(119899

1015840)otimesH(119877)

B11987110158401015840

(x10158401015840) (107)

Next we fix an arbitrary119898 ge 1 and modify the definitionof a PI ball

B(119873)

119871(x) is called PI iff diam (x) gt 119862

119873= 4119898119873

Therefore a PI ball admits a factorization B(119873)

119871(x) =

B1198711015840(x1015840) times B

11987110158401015840(x10158401015840) with dist(x1015840 x10158401015840) gt (119862

1198732119873)119871 = 2119898119871

so that with 119877119896= (119862

1198732119873)119871

119896 one has

120598 (119877119896) lt eminus2119898119871119896 lt

1

2eminus119871120573

119896 (108)

Lemma 32 Assume that the interaction U satisfies condition(U1) Fix an energy 119864 Let an 119873-particle ball B(119873)

119871119896(u) =

B(1198991015840)

119871119896(uJ)timesB

(11989910158401015840)

119871119896(uJ119888) and let a sample of the random potential

119881(sdot 120596) be such that

(a) dist(x1015840 x10158401015840) gt 119877119896= (119862

1198732119873)119871

119896

(b) B(1198991015840)

119871119896(u1015840) and B(119899

10158401015840)

119871119896(u10158401015840) are119898-Loc

(c) B(119873)

119871119896(u) is 119864-NR

Then the ball B(119873)

119871119896(u) is (119864119898)-NS

Proof Set 119877119896= (119862

1198732119873)119871

119896and define U(119877119896) and H(119877119896) as

above Denote

H(119877119896119873)

u119896 = H(119877119896)

B(119873)119871119896 (u)

G (119864) = (HB(119873)119871119896 (u)minus 119864)

minus1

(109)

and G(119877119896)(119864) = (H(119877119896119873)

u119896 minus 119864)minus1 The operator H(119877119896119873)

u119896 admitsthe decomposition

H(119877119896 119873)

u119896 = HB(1198991015840)119871119896

(u1015840) otimes 1(11989910158401015840)+ 1(119899

1015840)otimesHB(119899

1015840)119871119896

(x10158401015840) (110)

with u = (u1015840 u10158401015840) isin Z119889119899

1015840

times Z11988911989910158401015840

thus its eigenvalues are thesums119864

119886119887= 120582

119886+120583

119887 where 120582

119886 = Σ(HB119871119896 (u

1015840)) is the spectrum

ofHB119871119896 (u1015840)and respectively 120583

119887 = Σ(HB119871119896 (u

10158401015840)) Eigenvectors

of H(119877119896 119873)

u119896 can be chosen in the form Ψ119886119887= 120601

119886otimes 120595

119887 where

120601119886 are eigenvectors of HB(119899

1015840)119871119896

(u1015840) and 120595119887 are eigenvectors

of HB(11989910158401015840)119871119896

(u10158401015840) Note that for each eigenvalue 119864119886119887= 120582

119886+ 120583

119887

that is for each pair (120582119886 120583

119887) the nonresonance assumption


|119864 minus (120582119886+ 120583

119887)| ge e119871

120573

119896 reads as |(119864 minus 120582119886) minus 120583

119887)| ge e119871

120573

119896 and alsoas |(119864 minus 120583

119887) minus 120582

119886)| ge e119871

120573

119896 Therefore we can write

G(119877119896) (u y 119864)

= sum

120582119886

sum

120583119887

120601119886(u1015840)120601

119886(y1015840)120595

119887(u10158401015840)120595

119887(y10158401015840)

(120582119886+ 120583

119887) minus 119864

= sum

120582119886

P1015840

119886(u1015840 y1015840)G(119877119896)

B119871119896 (u10158401015840)(u10158401015840 y10158401015840 119864 minus 120582

119886)

= sum

120583119887

P10158401015840

119887(u10158401015840 y10158401015840)G(119877119896)

B119871119896 (u1015840)(u1015840 y1015840 119864 minus 120583

119887)

(111)

where the resolventsG(119877119896)

B119871119896 (u1015840)(119864minus120583

119887) andG(119877119896)

B119871119896 (u10158401015840)(119864minus120582

119886) are

nonresonant1003817100381710038171003817100381710038171003817G(119877119896)

B119871119896 (u1015840)(119864 minus 120583

119887)

1003817100381710038171003817100381710038171003817le e119871

120573

119896

1003817100381710038171003817100381710038171003817G(119877119896)

B119871119896 (u10158401015840)(119864 minus 120582

119886)

1003817100381710038171003817100381710038171003817le e119871

120573

119896

(112)

For any y isin 120597minusB119871119896(u) either 120588(u1015840

y1015840) = 119871119896 in which case we

infer from (b)

10038161003816100381610038161003816G(119877119896) (u y 119864)10038161003816100381610038161003816 le

10038161003816100381610038161003816B119871119896(u10158401015840)10038161003816100381610038161003816eminus120574(119898119871119896119873minus1)119871119896+119871

120573

119896 (113)

or 120588(u10158401015840 y10158401015840) = 119871

119896 and then we have

10038161003816100381610038161003816G(119877119896) (u y 119864)10038161003816100381610038161003816 le

10038161003816100381610038161003816B119871119896(u1015840)10038161003816100381610038161003816eminus120574(119898119871119896119873minus1)119871119896+119871

120573

119896 (114)

In either case the LHS is bounded by

exp (minus119898 (1 + 119871minus120591119896)119873minus(119873minus1)+1

119871119896+ 119871

120573

119896+ 119862 ln 119871

119896)

lt1

2eminus120574(119898119871119896119873)

(115)

Next using the second resolvent identity

G = G(119877119896) minus G(119877119896) (U minus U(119877119896))G (116)

and the assumed 119864-NR property of the resolvents G G(119877119896)we conclude that

10038171003817100381710038171003817G minus G(119877119896)

10038171003817100381710038171003817le10038171003817100381710038171003817U minus U(119877119896)

10038171003817100381710038171003817

10038171003817100381710038171003817G(119877119896)

10038171003817100381710038171003817G le eminus2119898119871119896e2119871

120573

119896

le1

2eminus120574(119898119871119896119873)

(117)

Now the claim follows from the inequality

10038161003816100381610038161003816G (x y 119864) minus G(119877119896) (x y 119864)10038161003816100381610038161003816 le

1

2eminus120574(119898119871119896 119873)

forallx y isin B(119873)

119871119896+1(u)

(118)

Lemma 33 Let B119871119896+1(x)B

119871119896+1(y) be two distant PI balls

Then for 1198710large enough

P B119871119896+1(x) B

119871119896+1(y) 119886119903119890 (119864119898) -119878

lt1

4119871minus119875(119873119896+1)

119896+1

(119)

Proof Let

R(2)= exist119864 B

119871119896+1(x) B

119871119896+1(y) 119886119903119890 119864-R

S(2)= B

119871119896+1(x) B119871119896+1

(y) 119886119903119890 (119864119898) -S (120)

ThenPS(2) le PR(2)

+PS(2)R(2)

andwithin the eventS(2)R(2) one of the balls B

119871119896+1(x) B

119871119896+1(y) must be 119864-NR

no matter how 119864 isin R is chosenNext consider the canonical factorization of the PI ball

B(119873)

119871119896+1(x) = B(119899

1015840)

119871119896+1(x1015840) times B(119899

1015840)

119871119896+1(x10158401015840) (121)

By Lemma 32 if both B119871119896+1(x1015840) and B

119871119896+1(x10158401015840) are119898-Loc and

B119871119896+1(x) is (119864119898)-S then B

119871119896+1(x) is 119864-R hence within the

complement of the event R(2) the other ball B119871119896+1(y) must

be 119864-NR On the other hand consider the canonical factor-ization of the PI ball

B(119873)

119871119896+1(y) = B(119899

1015840)

119871119896+1(y1015840) times B(119899

10158401015840)

119871119896+1(y10158401015840) (122)

If both B(1198991015840)119871119896+1(y1015840) and B(11989910158401015840)

119871119896+1(y10158401015840) are 119898-Loc and

B(119873)

119871119896+1(y) is (119864119898)-S then B(119873)

119871119896+1(y)must be 119864-R Let

L(4)= one of the balls B(119899

1015840)

119871119896+1(x1015840) B(119899

10158401015840)

119871119896+1(x10158401015840)

B(1198991015840)

119871119896+1(y1015840) B(119899

1015840)

119871119896+1(y10158401015840) is 119898-NL119900119888

(123)

Since these four balls correspond to systems with less than119873particles we can use induction in119873 and write

P L(4) le 4119871

minus119875(119873minus1119896+1)

119896+1= 4119871

minus2119875(119873119896+1)

119896+1 (124)

Then as we have noticed S(2)L(4)

sub R(2) hence for 1198710

large enough

P S(2) le P R

(2) + P L

(4)

le eminus119871120573

119896 + 4119871minus2119875(119873119896+1)

119896+1lt1

4119871minus119875(119873119896+1)

119896+1

(125)

The statement of Lemma 15 remains unchanged but itsproof requires a minor modification

Proof of Lemma 15 under hypothesis (U1) (A) By assumptioneither B

119871119896+1(u) is 119864-CNR and contains no (119864119898)-S ball of

radius 119871119896or there is a point w isin B

119871119896+1(u) such that any ball

B119871119896(k) sub B

119871119896+1(u)with 120588(w k) ge 119862

1198731198711+120575

119896is (119864119898)-NS In the

former case such an exclusion is unnecessary but in order to


treat both situations with one argument we can formally setw = u (or any other point)

Fix points x y with 119877 = 120588(x y) gt 1198711+984858119896

where 984858 = 2120575 isdefined in (103) Note that for 119871

0large (hence 119871

119896also large)

1198711+984858

119896≫ 119871

1+120575

119896 By triangle inequality

120588 (xB(119862119873minus1)119871119896

(w)) + 120588 (yB(119862119873minus1)119871119896(w))

ge 119877 minus (2119862119873minus 2) 119871

1+120575

119896

(126)

Assume first that

1199031015840= 120588 (xB

(119862119873minus1)1198711+120575119896(w)) ge 119871

119896+ 1

11990310158401015840= 120588 (yB

(119862119873minus1)1198711+120575119896(w)) ge 119871119896 + 1

(127)

All the balls of radius 119871119896both in B

1199031015840(x) and in B

11990310158401015840(y) are

automatically (119864119898)-NS Furthermore 1199031015840 + 11990310158401015840 ge 119877 minus 2(119862119873minus

1)1198711+120575

119896minus 2 ge 119877 minus 2119862

1198731198711+120575

119896



119891 B rarr C defined by 119891(x1015840 x10158401015840) = GB119871119896+1(x1015840 x10158401015840 119864)

Since 119864 is not a pole of the resolvent GB119871119896+1 (u)(sdot) it is well

defined (hence bounded on a finite set) By Lemma 10 119891 is(119871

119896 119902)-dominated both in x1015840 and in x10158401015840 with 119902 le eminus120574(119898119871119896119899)

Therefore one can write with the convention minusln 0 = +infinusing Lemma 7 and setting for brevity 119869 = 119872 minus 119899 + 1


ge minusln[(eminus119898(1+(12)119871minus120591119896 )119869119871119896)

(119877minus21198621198731198711+120575119896 minus2119871119896)(119871119896+1)

sdot e119871120573

119896+1] = 119898(1 +1

2119871minus120591

119896)

119869119871119896

119871119896+ 1119877(1 minus

3119862119873

119877

sdot 119871119896) minus 119871

120573

119896+1= 119898119877[(1 +

1

2119871minus120591

119896)

119869

(1 minus 119871minus1

119896)

sdot (1 minus3119862

119873119871119896

1198711+984858

119896

) minus119871120573

119896+1

119898119877] = 119898119877[(1 +

1

2119871minus120591

119896)

119869

sdot (1 minus 119871minus1

119896) (1 minus 3119862

119873119871minus984858

119896) minus

119871120573

119896+1

1198981198711+984858

119896

] ge 119898(1 +1

4

sdot 119871minus120591

119896)

119869

119877 ge 120574 (119898 119871119896+1) 119877

(128)


(B) This assertion is proved in the same way as itscounterpart in Section 33

The rest of the scaling procedure presented in Section 4requires no modification and applies to infinite-range inter-actions

8 Fermionic Hamiltonians onMore General Graphs

The reduction to a standard lattice Laplacian on a subset(119909

1 119909

119873) 119909

1lt sdot sdot sdot lt 119909

119873 with Dirichlet boundary

conditions is no longer possible for particle systems onlattices Z119889 with 119889 gt 1 Instead one has to work with asymmetric power of the lattice considered as a graph Soit seems reasonable to consider a fairly general countableconnected graph Z satisfying the condition of polynomialgrowth of balls

forall119909 isinZ forall119871 ge 11003816100381610038161003816119861119871 (119909)

1003816100381610038161003816 le 119862119889119871119889 (129)

where 119861119871(119909) = 119910 isin Z dZ(119909 119910) le 119871 In particular

this gives a uniform bound on the coordination numbers119899Z(119909) le 119862119889

(of course this bound may be nonoptimal)An orthonormal basis in the Hilbert space of square-

summable antisymmetric functionsΨ Z119873rarr C is formed

by the functions

Φa =1

radic119873

sum

120587isinS119873

119873

⨂

119895=1

1119886120587minus1(119895) (130)

where a = 1198861 119886

119873 with 119886

1 119886

119873 = 119873

Further define the graph (ZNgtEN

gt) as follows the vertex

set is

ZNgt

= a = 1198861 119886

119873 119886

119895isinZ 119886

1 119886

119873 = 119873

(131)

Two vertices a b form an edge iff

(i) the symmetric difference a ⊖ b has cardinality 1 thatis a = 119886

1 119888

2 119888

119873 b = 119887

1 119888

2 119888

119873 with

1198861 119887

1 119888

2 119888

119873 = 119873 + 1

(ii) dZ(1198861 1198871) = 1

Next define onZNgtthe max-distance

120588 (x y) = min120587isinS119873

max1le119895le119873

dZ (119909120587minus1(119895) 119910119895) (132)

and introduce the balls B119871(x) relative to the distance 120588(sdot sdot)

Now one can define the fermionic (negative) Laplacian(minusΔ) on ZN

gtand random Hamiltonians H(120596) = minusΔ +

119892V(x 120596) + U(x) where

U (x) = sum119894 =119895

119880(d (119909119894 119909

119895)) 119880 N 997888rarr R (133)

and the external random potential energy

V (x 120596) = 119881 (1199091 120596) + sdot sdot sdot + 119881 (119909

119873 120596) (134)

is generated by a random field 119881 Z times Ω rarr RThe method presented in Sections 3ndash7 applies to strongly

disordered random Hamiltonians H(120596) = minusΔ + 119892V(x 120596) +U(x) describing fermionic systems on connected graphs Z


with polynomial growth of balls indeed one can see thatwe did not use particular properties of the one-dimensionallattice Z = Z other than polynomially bounded growth ofballs

Strongly disordered bosonic systems can be treated in asimilar way the onlymodification required here concerns theexplicit form of the matrix elements of the Laplacian whichremains a second-order finite-difference operator

Appendix

A Proofs of Auxiliary Statements

A1 Proof of Lemma 17 We start with the second assertionThe random potential energy V(120596) reads as follows

V (1199091 119909

119873 120596) = sum

119910isinxn119910119881 (119910 120596) (A1)

Therefore if 120588(x y) = 0 then there exists a point119908 isin ΠB119871(u)

such that n119908(x) = n

119908(y) As a result

V (x 120596) minus V (y 120596) = (n119908(x) minus n

119908(y)) 119881 (119908 120596)

+ sum

V =119908

119888V119881 (V 120596) (A2)

where the explicit form of the integer coefficients

119888V = nV (x) minus nV (y) (A3)

is irrelevant for our argument it suffices to know that the sumin the RHS of (A2) is measurable with respect to the sigma-algebra F

=119908generated by the random variables 119881(V sdot) V =

119908 while

n119908(x) minus n

119908(y) = 119888

119908isin Z 0 hence 1003816100381610038161003816119888119908

1003816100381610038161003816 ge 1 (A4)

Therefore

P 1003816100381610038161003816119892V (x 120596) minus 119892V (y 120596)

1003816100381610038161003816 le 119904

= E [P 1003816100381610038161003816V (x 120596) minus V (y 120596)

1003816100381610038161003816 le10038161003816100381610038161198921003816100381610038161003816

minus1119904 | F

=119908]

= E [P 1003816100381610038161003816119888119908119881 (119908 120596) | z (120596)

1003816100381610038161003816 le10038161003816100381610038161198921003816100381610038161003816

minus1119904 | F

=119908]

(A5)

with some F=119908-measurable random variable z(120596) rendered

nonrandom by conditioning on F=119908 The event figuring in

the above conditional probability has the form

120596 119888119908119881 (119908 120596) isin [z (120596) minus

10038161003816100381610038161198921003816100381610038161003816

minus1119904 z (120596) +

10038161003816100381610038161198921003816100381610038161003816

minus1119904]

= 120596 119881 (119908 120596)

isin [z (120596) minus11990410038161003816100381610038161198921003816100381610038161003816

minus1

119888119908

z (120596) +11990410038161003816100381610038161198921003816100381610038161003816

minus1

119888119908

]

(A6)

Since |119888119908| ge 1 has length 2119904 we conclude that

P 1003816100381610038161003816V (x 120596) minus V (y 120596)

1003816100381610038161003816 le10038161003816100381610038161198921003816100381610038161003816

minus1119904

le sup119886isinR

(119865119881119908(119886 + 2119904) minus 119865

119881119908(119886))

(A7)

where 119865119881119908(sdot) is the conditional PDF of the random field 119881

at 119908 given F=119908 Since 119865

119881119908is continuous by assumption

(W1) the RHS of (A7) vanishes as |119892| rarr infin Therefore theprobability

P exist x y isin B119871(u) x = y 1003816100381610038161003816119892V (x) minus 119892V (y)

1003816100381610038161003816 le 119904 (A8)

tends to 0 as |119892| rarr infin so with arbitrarily high probabilitythe spectrum of the diagonal operatorV(120596) in B

1198710(u) admits

a positive uniform lower bound 119904 gt 0 on all spectralspacings (differences between the eigenvalues) By taking|119892| large enough all spacings for operator 119892V(120596) can bemade arbitrarily large Eigenvectors of a continuous finite-dimensional operator family 119860(119905) with simple spectrum at119905 = 119905

0are continuous in a neighborhood of 119905

0 To prove the

second assertion it suffices to apply this fact to the family119860(119905) =W minus 119892minus1119905Δ 119905 isin [0 1]

Theproof of the first assertion is even simplerUsing againrepresentation (A1) we see that for each x the value of thepotential energyV(x 120596) +U(x) is a linear combination (withinteger coefficients) of random variables with continuousprobability distribution obeying (W1) Arguing as above onecan see that for any 119864 isin R 119904 gt 0 and |119892| large enough withprobability arbitrarily close to 1 dist(119864 Σ(HB1198710 (u)

)) ge 119904 Bythe Combes-Thomas estimate [25] this implies exponentialdecay of the Green functions

100381610038161003816100381610038161003816GB1198710 (u)

(x y 119864)100381610038161003816100381610038161003816le eminus119898(119904)d(xy)

(A9)

with 119898(119904) rarr infin as 119904 rarr infin Since the graph distance d(sdot sdot)onZN

gtdominates the max-distance 120588(sdot sdot) the claim follows

A2 Proof of Lemma 20

Proof (A) Consider the set Π1198612119871(x) equiv ⋃

119899

119895=11198612119871(119909

119895) If

diamΠx gt 119860119873119871 this union cannot be connected for other-

wise it would have diameter le 119873 sdot 4119871 but diamΠ1198612119871(x) ge

diamΠx gt 119860119873119871 = 4119873119871 Therefore Π119861

1198711+120575(x) can be

decomposed into a disjoint union of two nonempty clusters

(⋃

119895isinJ

1198612119871(119909

119895))∐(⋃

119894isinJ119888

1198612119871(119909

119894)) (A10)

so for all 119895 isin J 119894 isin J119888 one has

120588 (119861119871(119909

119894) 119861

119871(119909

119895)) gt 2119871 (A11)

yielding assertion (B)(B) If 120588(x y) gt 119862

119873119871 then for some 119895

∘ 120588(119909

119895∘ 119910

119895∘) gt 119862

119873119871

Since both balls are FI for all 119895 isin [1 119899]

120588 (119909119895 119909

119895∘) le diamΠx le 119860

119873119871

120588 (119910119895 119910

119895∘) le diamΠy lt 119860

119873119871

(A12)

so by triangle inequality for all 119894 119895 isin [1 119899]

120588 (119909119894 119910

119895) ge 120588 (119909

119895∘ 119910

119895∘) minus 120588 (119909

119894 119909

119895∘) minus 120588 (119910

119895 119910

119895∘)

ge 119862119873119871 minus 2119860

119873119871 ge 3119871

(A13)


yielding

120588 (Π119861119871(x) Π119861

119871(y)) gt 3119871 minus 2119871 = 119871 (A14)

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

It is a pleasure to thank Tom Spencer Boris ShapiroAbel Klein and Misha Goldstein for fruitful discussions ofthe localization techniques the organizers of the programsldquoMathematics and Physics of Anderson Localization 50YearsAfterrdquo (2008) and ldquoPeriodic and Ergodic Spectral problemsrdquo(2015) at the Isaac Newton Institute Cambridge UK and theteam of the institute for their support and warm hospitalityand Shmuel Fishman Boris Shapiro and the Departmentof Physics of Technion Israel (2009) for their support andwarm hospitality

References

[1] P W Anderson ldquoAbsence of diffusion in certain random lat-ticesrdquo Physical Review vol 109 no 5 pp 1492ndash1505 1958

[2] D M Basko I L Aleiner and B L Altshuler ldquoMetal-insulatortransition in a weakly interacting many-electron system withlocalized single-particle statesrdquo Annals of Physics vol 321 no5 pp 1126ndash1205 2006

[3] I V Gornyi A D Mirlin and D G Polyakov ldquoInteractingelectrons in disordered wires anderson localization and low-T transportrdquo Physical Review Letters vol 95 Article ID 2066032005

[4] V Chulaevsky and Y Suhov ldquoMulti-particle Anderson local-isation induction on the number of particlesrdquo MathematicalPhysics Analysis and Geometry vol 12 no 2 pp 117ndash139 2009

[5] V Chulaevsky A Boutet de Monvel and Y Suhov ldquoDynamicallocalization for a multi-particle model with an alloy-typeexternal randompotentialrdquoNonlinearity vol 24 no 5 pp 1451ndash1472 2011

[6] M Aizenman and S Warzel ldquoLocalization bounds for multi-particle systemsrdquoCommunications inMathematical Physics vol290 no 3 pp 903ndash934 2009

[7] M Fauser and S Warzel ldquoMultiparticle localization for disor-dered systems on continuous space via the fractional momentmethodrdquo Reviews in Mathematical Physics vol 27 no 4 ArticleID 1550010 2015

[8] J Frohlich and T Spencer ldquoAbsence of diffusion in the Ander-son tight binding model for large disorder or low energyrdquoCommunications inMathematical Physics vol 88 no 2 pp 151ndash184 1983

[9] J Frohlich F Martinelli E Scoppola and T Spencer ldquoCon-structive proof of localization in the Anderson tight bindingmodelrdquo Communications in Mathematical Physics vol 101 no1 pp 21ndash46 1985

[10] H von Dreifus and A Klein ldquoA new proof of localizationin the Anderson tight binding modelrdquo Communications inMathematical Physics vol 124 no 2 pp 285ndash299 1989

[11] F Germinet and A Klein ldquoBootstrap multi-scale analysis andlocalization in randommediardquo Communications in Mathemati-cal Physics vol 222 no 2 pp 415ndash448 2001

[12] M Aizenman and S A Molchanov ldquoLocalization at large dis-order and at extreme energies an elementary derivationrdquoCommunications in Mathematical Physics vol 157 no 2 pp245ndash278 1993

[13] M Aizenman J H Schenker R M Friedrich and D Hundert-mark ldquoFinite-volume fractional-moment criteria for Andersonlocalizationrdquo Communications in Mathematical Physics vol224 no 1 pp 219ndash253 2001

[14] V Chulaevsky ldquoDirect scaling analysis of localization in single-particle quantum systems on graphs with diagonal disorderrdquoMathematical Physics Analysis and Geometry vol 15 no 4 pp361ndash399 2012

[15] Y G Sinai ldquoAnderson localization for one-dimensional dif-ference Schrodinger operator with quasiperiodic potentialrdquoJournal of Statistical Physics vol 46 no 5-6 pp 861ndash909 1987

[16] V Chulaevsky and Y Suhov ldquoEigenfunctions in a two-particleAnderson tight bindingmodelrdquo Communications inMathemat-ical Physics vol 289 no 2 pp 701ndash723 2009

[17] A Klein and S T Nguyen ldquoBootstrapmultiscale analysis for themulti-particle Anderson modelrdquo Journal of Statistical Physicsvol 151 no 5 pp 938ndash973 2013

[18] V Chulaevsky ldquoDirect Scaling Analysis of localizationin disordered systems II Multi-particle lattice systemsrdquohttparxivorgabs11062234

[19] V Chulaevsky and Y Suhov ldquoEfficient Anderson localizationboundsfor large multi-particle systemsrdquo Journal of SpectralThe-ory In press httpswwwems-phorgjournalsforthcomingphpjrn=jst

[20] V Chulaevsky ldquoOn resonances in disordered multi-particlesystemsrdquo Comptes Rendus de lrsquoAcademie des Sciences Series Ivol 350 pp 81ndash85 2011

[21] V Chulaevsky ldquoOn the regularity of the conditional distributionof the sample meanrdquo Markov Processes and Related Fields vol21 no 3 pp 415ndash431 2015

[22] V Chulaevsky and Y Suhov Multi-Scale Analysis for RandomQuantum Systems with Interaction Birkhaauser Boston MassUSA 2013

[23] M Aizenman A Elgart S Naboko J H Schenker and GStolz ldquoMoment analysis for localization in random Schrodingeroperatorsrdquo Inventiones Mathematicae vol 163 no 2 pp 343ndash413 2006

[24] T KatoPerturbationTheory for LinearOperators Springer NewYork NY USA 1976

[25] J M Combes and L Thomas ldquoAsymptotic behaviour of eigen-functions for multiparticle Schrodinger operatorsrdquo Communi-cations in Mathematical Physics vol 34 pp 251ndash270 1973

Submit your manuscripts athttpwwwhindawicom

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Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


12Motivation forThisWork The twomost popularmethodsof the rigorous Anderson localization theory the MultiscaleAnalysis developed in [8ndash11] and the Fractional MomentsMethod developed in [12 13] start with the analysis of theGreen functions (GFs) of a given random operator andthen one has to translate the upper bounds on the GFsobtained for a fixed energy (cf [8 11ndash13]) or in an energyinterval (cf [8 10]) into the language of the eigenfunctions(EFs) and eigenfunction correlators (EFCs) In a prior work[14] we proposed an alternative approach called the DirectScaling Analysis of eigenfunctions which focuses essentiallyon the eigenfunctions in finite domains of the configurationspace from the very beginning It keeps some importantadvantages of the MSA namely a greater tolerance than inFMM to lower regularity of the disorder and is close in spiritto the KAM (Kolmogorov-Arnold-Moser) approach whichoften requires a tedious inductive scheme In fact work [14]has been inspired by the KAM approach to quasi-periodicHamiltonians proposed earlier by Sinai [15] Compared to theKAM method the DSA has a simpler structure of the scaleinduction similar to that of theMSA although it also inheritssomewhat weaker probabilistic bounds typical of the MSA

Furthermore as was said there are two kinds of theMSAwith a fixed energy (FEMSA) and with a variable energy(VEMSA) the latter is often referenced to as the energy-interval MSA for it treats the decay properties of the Greenfunctions in an entire interval of energies at onceTheFEMSAis slightly simpler than VEMSA in the single-particle casebut in the realm of multiparticle random Hamiltonians witha nontrivial interaction the variable-energy variant of the(multiparticle) MSA is considerably more complex Some ofthe components of the VEMPMSA are simply eliminatedwith the (MP)DSA approach and the counterparts of someother components become quite elementary A good exampleof such kind is Lemma 21 (see Section 43) the proof ofwhich is very simple and fits in a few lines while its analogin the variable-energy MPMSA (cf eg [4 Lemma 3]) issubstantially more involved

This constatation has been the principal motivation forthe present work A reader familiar with the techniques ofthe papers [4 16 17] can see that the multiparticle variantof the DSA (MPDSA) from the present paper is much closerin its logical structure to the FEMSA yet it leads directly tothe VEMSA-type decay bounds on the entire eigenbases inarbitrarily large finite domains

In preprint [18] which has been a starting point for thepresent paper we gave the first proof of exponential decay ofmultiparticle eigenfunctions in presence of an infinite-rangeinteraction decaying at infinity at a fractional-exponentialrate 119903 997891rarr eminus119903

120577

0 lt 120577 lt 1 In a recent work [7] Aizenmanand Warzel further developing the techniques of [6] provedfractional-exponential decay of EF correlators (EFCs) forsubexponentially decaying interactions of infinite range Dueto the logical structure of the FMM the decay analysis ofthe eigenfunctions is subordinate to that of the EFCs andthis explains why [7] established only a subexponential decayof eigenfunctions The DSA is free from this limitation andthis allows us to prove a genuine exponential decay of the

multiparticle EFs even in the case where the interaction decayis subexponential

13 Novelty Apart from simplifications of the MultiparticleMultiscale Analysis (MPMSA) developed in [4] the noveltyof this paper is that the external random potential is notnecessarily IID (with independent and identically distributedvalues) but is strongly mixing Earlier publications including[19] required the common external random potential actingon the particles to be IID

14 Structure of the Paper The structure of the paper is asfollows

(i) The principal assumptions andmain results are statedin Sections 24ndash26

(ii) The deterministic techniques of scaling analysis arepresented in Section 3

(iii) In Section 4 we treat first the case of finite-rangeinteractions in order to present the logical structureand the methodological advantages of the MPDSA inthe most transparent way

(iv) In Section 7 our analysis is adapted to the interac-tions of infinite range

(v) Sections 5 and 6 are devoted to the derivation ofspectral and dynamical localization from the finalresults of the MPDSA

2 Preliminaries Assumptionsand Main Results

21 Configurations of Indistinguishable Particles in Z1 In thefirst part of this paper we work with configurations of 119873 ge1 quantum particles in the one-dimensional lattice Z =

Z1 In quantum mechanics particles of the same kind areconsidered indistinguishable more precisely depending onthe nature of the particles the wave functions describing119873 gt 1 particles have to be either symmetric (Bose-Einsteinquantum statistics) or antisymmetric (Fermi-Dirac quantumstatistics) We choose here the fermionic case this gives riseto slightly simpler notations and constructions

For clarity we use often boldface symbols for the objectsrelated to the multiparticle systems

Quantum states of an 119873-particle fermionic system in Z

are elements of the Hilbert spaceHN=H119873minus formed by all

square-summable antisymmetric functions Ψ Z119873rarr C

with the inner product ⟨sdot | sdot⟩ inherited from the Hilbertspace ℓ2(Z119873

) = (H1)otimes119873 In this particular case where

the ldquophysicalrdquo configuration space is one-dimensional H119873

admits a simple representation which we will useFirst note that any antisymmetric function

x = (1199091 119909

119873) 997891997888rarr Ψ (119909

1 119909

119873) (3)

vanishes on all hyperplanes x 119909119894= 119909

119895 1 le 119894 lt 119895 le 119873

Further any function defined on the ldquopositive sectorrdquoZNgt=

(1199091 119909

119873) 119909

1gt 119909

2gt sdot sdot sdot gt 119909

119873 admits a unique




Φa (x) =1

radic119873

sum

120587isinS119873

(minus1)120587

119873

⨂

119895=1

1119886120587minus1(119895) (4)


119873 = 119873 and 1




120587minus1(1) 119886



gt) of square-




119873


ge exist119894 = 119895 119909

119894=




119886119873 a isinZN

gt (5)


119873




(ΔG119891) (119909) = sum

⟨119909119910⟩

(119891 (119910) minus 119891 (119909))

= minus119899G (119909) 119891 (119909) + sum

⟨119909119910⟩

119891 (119910)

(6)



1= |119909

1minus 119910


119873minus 119910




Δ =

119873

sum

119895=1

(

119895minus1

⨂

119894=1

1(119894)) otimes Δ(119895) otimes (119873

⨂

119896=119895+1

1(119896)) (7)


(Δ(119895)119891) (119909

119895) = minus2119891 (119909

119895) + 119891 (119909

119895minus 1) + 119891 (119909

119895+ 1)

119909119895isin Z

(8)





119895minus 119887

119895| = 1 while for all 119894 = 119895 119886

119894= 119887

119894



restriction to ZNgt







defined by

120588 (x y) = max1le119895le119873

dZ (119909119895 119910119895) (9)


B119871(x) = y 120588 (x y) le 119871 =

119873

⨉

119895=1

119861119871(119909

119895) (10)

Here 119861119871(119909) = [119909 minus 119871 119909 + 119871] cap Z





119899


gtand x10158401015840 isinZ(n10158401015840)

gtin a usual way

dist (x1015840 x10158401015840) = min119906isinx1015840



x1015840 isin Z(n1015840)gt

and x10158401015840 isin Z(n10158401015840)gt


B(119873)

119871(x) = B(119899

1015840)

119871(x1015840) times B(119899

10158401015840)

119871(x10158401015840) (12)









gt(yΛ119888) = 1

120597+Λ = 120597

minusΛ

119888


(13)


1 119909


nx(119910) = 119895 119909119895= 119910 119910 isin X


gt letΣ(H


Λ

andGΛ(119864) = (H

Λminus119864)







H (120596) = H0+W (14)




119892V (x 120596) = 119892119881 (1199091 120596) + sdot sdot sdot + 119892119881 (119909

119873 120596) (16)



U (x) = sum119894 =119895

119880(10038161003816100381610038161003816119909119894minus 119909

119895

10038161003816100381610038161003816) (17)


0 lt infin




tions




119865119881119909(119905) = P 119881 (119909 120596) le 119905 119909 isinZ (18)


119865119881119909(119905 | F

=119909) = P 119881 (119909 120596) le 119905 | F

=119909 (19)





sup119909isinZ

sup119886isinR

[119865119881119909(119886 + 119904 | F

119881

=119909) minus 119865

119881119909(119886 | F

119881

=119909)] le 119862119904 (20)



Λthe



1015840 119861

10158401015840) ge 119871 ge 1 any events E1015840

isin

F119881

1198611015840 E10158401015840

isin F119881

11986110158401015840 and some 119862 gt 0

10038161003816100381610038161003816P E

1015840capE

10158401015840 minus P E

1015840P E

1015840101584010038161003816100381610038161003816le eminus119862 ln2119871

(21)



eminus119862 ln2(1198710)120572119896

lt ((1198710)120572119896

)

minus119886119901(1+120579)119896

119896 ge 0 (22)








120585119876(120596) = |119876|

minus1sum

119909isin119876

119881 (119909 120596) (23)


119876(120596)119909 isin 119876 Denote byF


generated by 120578119909 119881

119910 119909 isin 119876 119910 notin 119876


(W3) There exist1198621015840 119860

1015840 119887




P 120585119876(120596) isin [120582 (120596) 120582 (120596) + 119904] le 119862

1015840|119876|

1198601015840

1199041198871015840

(24)









1lowast 120583

2 where 120583

2is



0 lt 120588lowastle 120588

1 (119905) le 120588 lt infin

1205881015840

1(119905)

1205881(119905)le 119862

1lt infin

forall119905 isin (119886 119887)

(25)




119880 (119903) = 0 forall119903 ge 1199030 (26)


|119880 (119903)| le 119862eminus1198881199031minus120577

forall119903 ge 0 (27)


26 Main Results








119895 x)

large enough

10038161003816100381610038161003816Ψ

119895(x 120596)10038161003816100381610038161003816 le eminus119898120588(x119895x) 119898 gt 0 (28)


E[sup119905isinR

10038161003816100381610038161003816⟨1y10038161003816100381610038161003816eminusi119905H(120596)10038161003816100381610038161003816

1x⟩10038161003816100381610038161003816] le 119862eminus119886 ln

1+119888120588(xy) (29)


E[sup119905isinR

100381710038171003817100381710038171003817e(1198862) ln








H(120596) on 119868




119873

y = 1199101 119910

119873


dZ (119909119894 y) max119894

dZ (119910119894 x)] (31)

where

dZ (119909 y) = min119895

dZ (119909 119910119895) (32)


E[sup119905isinR

10038161003816100381610038161003816⟨1y10038161003816100381610038161003816eminusi119905H(120596)10038161003816100381610038161003816

1x⟩10038161003816100381610038161003816] le 119862 (x) eminus119886 ln

1+119888dH(xy) (33)



119871(119904) le 119862119871

119860119904119899 119862119860 isin

(0 +infin) and 119887 isin (0 1]





gt |Λ| equiv


HΛ (120596) = 1

ΛH (120596) 1Λ HΛ (34)







(Γxy119891) (x) = 120575xy119891 (y) (35)


Λ(119864) = (H

Λminus119864)




10038161003816100381610038161003816

le 119862ℓ


10038161003816100381610038161003816GBℓ(x) (x k)

10038161003816100381610038161003816max

k1015840isin120597+Bℓ(x)

10038161003816100381610038161003816GB119871 (k

1015840 y)10038161003816100381610038161003816

(36)

where

119862ℓ=1003816100381610038161003816120597Bℓ

(k)1003816100381610038161003816 le 119862 (119873) ℓ119873 (37)


10038161003816100381610038161003816G

119861119871(x y)10038161003816100381610038161003816

le (119862ℓmax

120588(xk)=ℓ

10038161003816100381610038161003816GBℓ(x) (x k)

10038161003816100381610038161003816) max

120588(uk1015840)leℓ+1

10038161003816100381610038161003816GB119871 (k

1015840 y)10038161003816100381610038161003816

(38)


10038161003816100381610038161003816GB119871 (x y 119864)

10038161003816100381610038161003816

le 119862ℓ

10038171003817100381710038171003817GBℓ(u) (119864)

10038171003817100381710038171003817max

k120588(uk)leℓ+1

10038161003816100381610038161003816GB119871 (k y 119864)

10038161003816100381610038161003816

(39)


ℓ(u)

1003816100381610038161003816120595 (x)1003816100381610038161003816 le 119862ℓ

10038171003817100381710038171003817GBℓ(x) (119864)

10038171003817100381710038171003817max

y120588(xy)leℓ+11003816100381610038161003816120595 (y)

1003816100381610038161003816 (40)



gt


119877(119906) rarr R



ℓ(119909) sub Λ

119877(119906) one has

119891 (119909) le 119902 max119910d(119909119910)leℓ

119891 (119910) (41)





M (119891 119861)

le 119902(119871minusℓ)(ℓ+1)

M (119891 119861)

M (119891 Λ) = max119909isinΛ

1003816100381610038161003816119891 (119909)1003816100381610038161003816

(42)





M (119891 119861)

(43)





1015840) ⊊ Λ and in 11990910158401015840 isin 119861

11990310158401015840(119906

10158401015840) ⊊ Λ with

d(1199061015840 11990610158401015840) ge 1199031015840 + 11990310158401015840 + 2 Then

119891 (1199061015840 119906

10158401015840) le 119902


10158401015840+1)(ℓ+1)rfloor

M (119891 119861)

le 119902(1199031015840+11990310158401015840minus2ℓ)(ℓ+1)

M (119891 119861)

(44)



tion

11989111991010158401015840 119910

1015840997891997888rarr 119891 (119910

1015840 119910

10158401015840) (45)


1015840) ⊊ Λ so by Lemma 6

11989111991010158401015840 (119906

1015840) = 119891 (119906

1015840 119910

10158401015840) le 119902

lfloor(1199031015840+1)(ℓ+1)rfloor

M (119891 Λ times Λ) (46)


10158401015840997891rarr 119891(119906

1015840 119910

10158401015840) which is

(ℓ 119902)-dominated in 11986111990310158401015840(119906






119896+1=

lceil119871120572

119896rceil 119871



0to be large enough


119871(u) is called




ℓ(u) sube B

119871(u) with ℓ ge 1198711120572





10038161003816100381610038161003816GB119871(u) (u y 119864 120596)

10038161003816100381610038161003816le eminus120574(119898119871)119871+2119871

120573

(47)

where

120574 (119898 119871) = 119898 (1 + 119871minus120591) 120591 =

1

8 (48)




120595119895 119895 = 1 |B



0isin B

119871119896(u)withd(x

0 y

0) gt 119871

119896minus1+1

and an integer

119877 isin [119871119896minus1 d (x

0 y

0) minus 119871

119896minus1+ 1] (49)


119877(x

0) is (119864119898)-NS and set

119902 = eminus120574(119898ℓ)ℓ (50)

Then



120595119895=

|120595119895⟩⟨120595





B119877(x

0) ni x 997891997888rarr

100381610038161003816100381610038161003816GB119871119896 (u)

(x y 119864)100381610038161003816100381610038161003816

(51)


0) and globally



119877(y


0fixed




satisfies10038161003816100381610038161003816120595

119895(x)120595

119895(y)10038161003816100381610038161003816 le eminus120574(119898119871)120588(xy) (52)




120588(x y) ge 119862119873119871 with 119862

119873= 2119860

119873+ 3 119860

119873= 4119873



in Section 43)



ℓ(y) and119898-







1003816100381610038161003816GB (x y 119864)1003816100381610038161003816 le sum

119864119895isinΣ(HB)

10038161003816100381610038161003816120595

119895(x)1003816100381610038161003816100381610038161003816100381610038161003816120595

119895(y)10038161003816100381610038161003816

10038161003816100381610038161003816119864 minus 119864

119895

10038161003816100381610038161003816

(53)


120573


1003816100381610038161003816GB (x y 119864)1003816100381610038161003816 le eminus120574(119898119871)119871+119871

120573+ln|B|

le eminus120574(119898119871)119871+2119871120573

(54)



119898(1 + 119871minus120591

119896) 119871

119896minus 2119871

120573

119896= 119898119871

119896+ 119898119871

1minus120591

119896minus 2119871

120573

119896

ge 119898(1 +1

2119871minus120591

119896) 119871

119896

(55)




(1)




(119864119898)-NS(B) if a ball B


(119864119898)-NS




119871119896(k) sub B

119871119896(u) with






120588 (xB(119862119873minus1)119871119896

(w)) + 120588 (yB(119862119873minus1)119871119896(w))

ge 119877 minus (2119862119873minus 2) 119871

119896

(56)

Assume first that

1199031015840= 120588 (xB

(119862119873minus1)119871119896(w)) ge 119871

119896+ 1

11990310158401015840= 120588 (yB

(119862119873minus1)119871119896(w)) ge 119871

119896+ 1

(57)



119896are auto-


Furthermore

1199031015840+ 119903

10158401015840ge 119877 minus 2 (119862

119873minus 1) 119871

119896minus 2 ge 119877 minus 2119862

119873119871119896 (58)










(119877minus2119862119873119871119896minus2119871119896)(119871119896+1)

e119871120573

119896+1]

= 119898(1 +1

2119871minus120591

119896)119871119896

119871119896+ 1119877 (1 minus 3C

119873119877minus1119871119896) minus 119871

120573

119896+1

= 119898119877[(1 +1

2119871minus120591

119896) (1 minus 119871

minus1

119896) (1 minus 3119862

119873119871minus984858

119896)

minus119871120573

119896+1

119898119877] ge 119898(1 +

1

4119871minus120591

119896)119877 ge 120574 (119898 119871

119896+1) 119877

(59)




B119871119896(x) B


119871119896+1(u) is 119864-


119871119896(y) must


the hypothesis






119895 (k1015840 k10158401015840) 997891rarr |120595

119895(k1015840)120595

119895(k10158401015840)| where

120595119895 119895 isin [1 |B





119895 = 1 so the


119871119896+1(u) and

assume 120588(x y) = 119877 ge 1198711+984858119896




minus ln 10038161003816100381610038161003816120595119895 (x)120595119895 (y)10038161003816100381610038161003816


(119877minus2119862119873119871119896minus2119871119896)(119871119896+1)

]

(60)






120588 (x y) ge 1198711+984858119896minus1

0 lt 120591 lt 120588

1 + 984858 lt 120572

120573 lt 1 minus 120591

(61)


120572 =4

3

120588 =1

6

120591 =1

8

120573 =1

2

(62)




119901 gt 0 there is 1198920= 119892

0(119898 119901 119871


|119892| ge 1198920 any ball B

1198710(119906) and any 119864 isin R

P B1198710(u) 119894119904 (119864119898) -119878 le 119871minus119901

0

P B1198710(u) 119894119904 119898-NL119900119888 le 119871

minus119901

0

(63)




1015840isin (0 1] 1198601015840


119871(u1015840) B(119873)



P dist (Σ119871u1015840 Σ119871u10158401015840) le 119904 le (3119871)

2119873ℎ119871 (2119904) (64)


HB119871(u10158401015840) and

ℎ119871 (119904) = 119862

1015840101584011987111986010158401015840

119904119887and1198871015840

119887 and 1198871015840equiv min [119887 1198871015840] (65)

and 11986010158401015840= 119860

10158401015840(119860



u10158401015840) gt 4119873119871

P exist119864 isin R B119871(u1015840) B

119871(u10158401015840) 119886119903119890 119864-119875119877 le eminus119871

1205731015840

(66)





119873= 4119873 and





H(119873)

B119871(u) = H(1198991015840)

B119871(u1015840)otimes 1(119899

10158401015840)+ 1(119899

1015840)otimesH(119899

10158401015840)

B119871(u10158401015840)(67)

with 1198991015840+11989910158401015840 = 119873 u = (u1015840 u10158401015840) u1015840isinZn1015840

gt u10158401015840isinZn10158401015840

gt while for




(x) ΠJ119888B119871(x)) gt 2119871

(B) If B119871(x) B


then 120588(ΠB119871(x) ΠB

119871(y)) ge 2119871






B(119873)

119871119896(u) = B(119899

1015840)

119871119896(u1015840) times B(119899

10158401015840)

119871119896(u10158401015840) (68)


If the balls B(1198991015840)

119871119896minus1(u1015840) and B(119899

10158401015840)


B(119873)

119871119896(u) is also119898-Loc



HB119871119896 (u)= HB(119899

1015840)119871119896

(u1015840) otimes 1(11989910158401015840)+ 1(119899

1015840)otimesHB(119899

10158401015840)119871119896

(u10158401015840) (69)



119895= 120601

1198951015840 otimes120595

11989510158401015840 where 120601


1015840)119871119896



10158401015840)119871119896

(u10158401015840) Now


119895 (x)Ψ119895 (y)10038161003816100381610038161003816

=100381610038161003816100381610038161206011198951015840 (x1015840)120601

11989510158401015840 (x10158401015840)120595

1198951015840 (y1015840)120595

11989510158401015840 (y10158401015840)10038161003816100381610038161003816

(70)


1015840)

119871119896(u1015840) and B(119899

10158401015840)

119871119896(u10158401015840)


119875 (119899 119896) = 2119873minus119899119901 (1 + 120579)

119896 1 le 119899 le 119873 119896 ge 0 (71)



119871119896(u)

P B(119899)

119871119896(u) 119894119904 119898-NL119900119888 le 119871

minus119875(119899119896)

119896 (72)


119871119896(u) one has

P B(119873)

119871119896(u) 119894119904 119898-NL119900119888 le 2119871

minus2119875(119873119896)

119896le1

4119871minus119875(119873119896)

119896 (73)




1015840)

119871119896(u1015840) B(119899

10158401015840)

119871119896(u10158401015840) is 119898-NLoc


P B(119873)

119871119896(u) is 119898-NL119900119888

le P B(11989910158401015840)

119871119896(u10158401015840) is 119898-NL119900119888

+ P B(11989910158401015840)

119871119896(u10158401015840) is 119898-NL119900119888

le 119871minus119875(1198991015840119896)

119896+ 119871

minus119875(11989910158401015840119896)

119896le 2119871

minus2119875(119873119896)

119896lt1

4119871minus119875(119873119896)

119896

(74)





1198961015840 with 1198961015840 le 119896


119871119896(u)

P B(119873)

119871 119894(u) 119894119904 119898-NL119900119888 le 119871

minus119875(119873119894)

119894 (75)


119871119896+1(x) one has

P B(119873)

119871119896+1(u) 119894119904 119898-NL119900119888 le 119871

minus119875(119873119896+1)

119896+1 (76)

provided that

119901 gt2120572

2

2 minus 1205722119873119889

0 lt 3120579 le min2 minus 1205722

1205722minus2119873119889


(77)


119871119896(u) B(119873)

119871119896(k) one

has for some 119886 119888 gt 0


119871119896(u) B(119873)

119871119896(k) 119886119903119890 (119864119898) -119878

le 119871minus119875(119873119896)

119896equiv 119871

minus119901(1+120599)119896


1+119888119871119896

(78)


N119896+1= B(119873)

119871119896+1(u) is 119898-NLo119888

S(2)

119896


119871119896(u) B(119873)

119871119896(k)

sub B(119873)

119871119896+1(w)

R(2)


119871119896(u) B(119873)

119871119896(k)

sub B(119873)

119871119896+1(w)

(79)


N119896+1sub S

(2)

119896subR

(2)

119896cup (S

(2)

119896R

(2)

119896) (80)


119896 le eminus119871

1205731015840


119896R

(2)


119871119896+1(w) and


S(2)

119896(u k) = exist119864 B(119873)

119871119896(u) B(119873)

119871119896(k) are (119864119898) -S (81)


119896(u k) R(2)

119896 either B(119873)

119871119896(u) or B(119873)

119871119896(k)


119871119896(u) is 119864-CNR



pair of 119898-NL119900119888 balls B(119873)

119871119896minus1(y1015840) B(119873)


(12)1198711+120575


(1) Both B(119873)

119871119896minus1(y1015840) and B(119873)

119871119896minus1(y10158401015840) are FI


120588 (ΠB119871119896minus1(u) ΠB

119871119896minus1(k)) ge 2119871

119896 (82)


119896minus1= 119871

11205722

119896+1and 119871

0large enough

P B(119873)

119871119896minus1(y1015840) B(119873)

119871119896minus1(y10158401015840) are 119898-NL119900119888

le 2119871minus(2119901120572

2)(1+120579)

119896minus1

119896+1

(83)


119871119896(w)|2 le (321198732)1198712119873


P S(2)

119896R

(2)

119896 le32119873

2119871minus(2119901120572

2)(1+120579)

119896minus1+2119873

119896+1 (84)



119896+1 under con-


2119901

12057222119873minus119873119901 (1 + 120579)

119896minus1minus 2119873119889 gt 2

119873minus119873119901 (1 + 120579)

119896+1 (85)

or equivalently

(1 + 120579)2lt2119901

1205722minus

2119873119889

119901 (1 + 120579)119896minus1 (86)


3120599 le2 minus 120572

2

1205722minus2119873

119901 (87)


P S(2)

119896 le P R

(2)

119896 + P S

(2)

119896R

(2)

119896

le eminus1198711205731015840

119896 +1

4119871minus119875(119873119896+1)

119896+1lt1

2119871minus119875(119873119896+1)

119896+1

(88)


119871119896minus1(y1015840) orB(119873)


P B(119873)

119871119896minus1(y1015840) B(119873)

119871119896minus1(y10158401015840) is 119898-NL119900119888

le min (P B(119873)

119871119896minus1(y1015840) is 119898-NL119900119888

P B(119873)

119871119896minus1(y10158401015840) is 119898-NL119900119888) le 119871

minus2119901(1+120579)119896minus1

119896minus1

lt 119871minus119901(1+120579)

119896+1

119896+1

(89)



119871minus119875(119873119896)

119896equiv 119871

minus119901(1+120599)119896


1+119888119871119896 (90)




P B(119873)

119871119896(u) 119894119904 119898-NL119900119888 le

1

4119871minus119901(1+120579)

119896

119896(91)


minus119901(1+120579)119896


119895of operator HB(119873)119871119896 (u)



119896one

has10038161003816100381610038161003816Ψ119895(x) Ψ






1003816100381610038161003816Ψ (y)1003816100381610038161003816 le eminus119898y

(93)








119871119896(y) sub

B119871119896+2(z) Fix 120596 isin Ω1015840


2le 1 implies Ψ

infinle 1 there is


119899isin B

119871119896minus1(0) then






119871119896+2(z) with 120588(y z) isin [119871

119896+1 119871

119896+2)


1+120575

119896) is (120582119898)-NS Note that



119877(y) with

119877 = 120588(y z)minus21198711+120575119896minus1 gt (12)119871

1+2120575



minusln 1003816100381610038161003816Ψ (y)

1003816100381610038161003816

120588 (y z)ge 119898 (1 + 119871

minus120591

119896) (1 minus

2119871119896+ 1

120588 (y z))

ge 119898(1 +1

2119871minus120591

119896) gt 119898

(95)









119871(119909) B


P exist 119864 isin 119868 B119871(x) B

119871(y) 119886119903119890 (119864119898) -S le 119891 (119871) (96)


119871(119909) cup B

119871(119910) one has

E[ sup120601isinB1(119868)

10038161003816100381610038161003816⟨1x1003816100381610038161003816120601 (HΛ (120596))

1003816100381610038161003816 1y⟩10038161003816100381610038161003816]

le 119862119871119889119890minus119898119871+ 119891 (119871)

(97)




Λ sup B119871(119909) cup B

119871(119910)


10038161003816100381610038161003816⟨1x1003816100381610038161003816120601 (HΛ (120596))

1003816100381610038161003816 1y⟩10038161003816100381610038161003816] le 119862119890

minus119886 ln1+119888120588(xy) (98)










same 119888 119886 gt 0 as in (98)

E[ sup120601isinC1(119868)

10038161003816100381610038161003816⟨1x1003816100381610038161003816120601 (H (120596))

1003816100381610038161003816 1y⟩10038161003816100381610038161003816] le 119862eminus119886 ln

1+119888120588(xy) (99)




int120601 (120582) 119889120583xyB120596 (120582)

= ⟨1x1003816100381610038161003816 120601 (HB (120596))Π119868

(HB (120596))100381610038161003816100381610038161y⟩

(100)








119899rarr H with







119871119896

E [1003816100381610038161003816120583

xy120596


E [100381610038161003816100381610038161003816120583xyB119871119896

100381610038161003816100381610038161003816(E)] (101)




120601119905 120582 997891997888rarr 119891

119868(120582) ei119905120582 119905 isin R (102)








984858 = 2120575

120572 = 1 + 4120575

120591 =120575

2

(103)



120574 (119898 119871 119899) = 119898 (1 + 119871minus120591)119873minus119899+1

(104)



120588(x y) ge 1198621198731198711+120575 with 119862

119873= 4119873






119871(x) admits

a factorization

B(119873)

119871(x) = B(119899

1015840)

119871(x1015840) times B(119899

10158401015840)

119871(x10158401015840) (105)


119871(x) B


then 120588(ΠB119871(x) ΠB

119871(y)) ge 1198711+120575




119877119877 ge 119903



119880(2)

119877(119903) = 1

119903le119877119880

(2)(119903) (106)


gt



119877(x1015840) + U



11987110158401015840(x10158401015840) with

120588(B1198711015840(x1015840)B



H(119877)

B = H(119877)

B1198711015840(x1015840) otimes 1

(11989910158401015840)+ 1(119899

1015840)otimesH(119877)

B11987110158401015840

(x10158401015840) (107)


B(119873)


119873= 4119898119873


119871(x) =

B1198711015840(x1015840) times B

11987110158401015840(x10158401015840) with dist(x1015840 x10158401015840) gt (119862

1198732119873)119871 = 2119898119871

so that with 119877119896= (119862

1198732119873)119871

119896 one has

120598 (119877119896) lt eminus2119898119871119896 lt

1

2eminus119871120573

119896 (108)


119871119896(u) =

B(1198991015840)

119871119896(uJ)timesB

(11989910158401015840)



(a) dist(x1015840 x10158401015840) gt 119877119896= (119862

1198732119873)119871

119896

(b) B(1198991015840)

119871119896(u1015840) and B(119899

10158401015840)

119871119896(u10158401015840) are119898-Loc

(c) B(119873)

119871119896(u) is 119864-NR


119871119896(u) is (119864119898)-NS

Proof Set 119877119896= (119862

1198732119873)119871


above Denote

H(119877119896119873)

u119896 = H(119877119896)

B(119873)119871119896 (u)

G (119864) = (HB(119873)119871119896 (u)minus 119864)

minus1

(109)

and G(119877119896)(119864) = (H(119877119896119873)



H(119877119896 119873)

u119896 = HB(1198991015840)119871119896

(u1015840) otimes 1(11989910158401015840)+ 1(119899

1015840)otimesHB(119899

1015840)119871119896

(x10158401015840) (110)

with u = (u1015840 u10158401015840) isin Z119889119899

1015840

times Z11988911989910158401015840


119886119887= 120582

119886+120583

119887 where 120582

119886 = Σ(HB119871119896 (u



119887 = Σ(HB119871119896 (u


of H(119877119896 119873)


119886otimes 120595

119887 where


1015840)119871119896


of HB(11989910158401015840)119871119896


119886+ 120583

119887




|119864 minus (120582119886+ 120583

119887)| ge e119871

120573


119887)| ge e119871

120573

119896 and alsoas |(119864 minus 120583

119887) minus 120582

119886)| ge e119871

120573


G(119877119896) (u y 119864)

= sum

120582119886

sum

120583119887

120601119886(u1015840)120601

119886(y1015840)120595

119887(u10158401015840)120595

119887(y10158401015840)

(120582119886+ 120583

119887) minus 119864

= sum

120582119886

P1015840

119886(u1015840 y1015840)G(119877119896)

B119871119896 (u10158401015840)(u10158401015840 y10158401015840 119864 minus 120582

119886)

= sum

120583119887

P10158401015840

119887(u10158401015840 y10158401015840)G(119877119896)

B119871119896 (u1015840)(u1015840 y1015840 119864 minus 120583

119887)

(111)


B119871119896 (u1015840)(119864minus120583

119887) andG(119877119896)

B119871119896 (u10158401015840)(119864minus120582

119886) are

nonresonant1003817100381710038171003817100381710038171003817G(119877119896)

B119871119896 (u1015840)(119864 minus 120583

119887)

1003817100381710038171003817100381710038171003817le e119871

120573

119896

1003817100381710038171003817100381710038171003817G(119877119896)

B119871119896 (u10158401015840)(119864 minus 120582

119886)

1003817100381710038171003817100381710038171003817le e119871

120573

119896

(112)


y1015840) = 119871119896 in which case we

infer from (b)

10038161003816100381610038161003816G(119877119896) (u y 119864)10038161003816100381610038161003816 le

10038161003816100381610038161003816B119871119896(u10158401015840)10038161003816100381610038161003816eminus120574(119898119871119896119873minus1)119871119896+119871

120573

119896 (113)

or 120588(u10158401015840 y10158401015840) = 119871


10038161003816100381610038161003816G(119877119896) (u y 119864)10038161003816100381610038161003816 le

10038161003816100381610038161003816B119871119896(u1015840)10038161003816100381610038161003816eminus120574(119898119871119896119873minus1)119871119896+119871

120573

119896 (114)



119871119896+ 119871

120573

119896+ 119862 ln 119871

119896)

lt1

2eminus120574(119898119871119896119873)

(115)




10038171003817100381710038171003817G minus G(119877119896)

10038171003817100381710038171003817le10038171003817100381710038171003817U minus U(119877119896)

10038171003817100381710038171003817

10038171003817100381710038171003817G(119877119896)

10038171003817100381710038171003817G le eminus2119898119871119896e2119871

120573

119896

le1

2eminus120574(119898119871119896119873)

(117)


10038161003816100381610038161003816G (x y 119864) minus G(119877119896) (x y 119864)10038161003816100381610038161003816 le

1

2eminus120574(119898119871119896 119873)


119871119896+1(u)

(118)

Lemma 33 Let B119871119896+1(x)B



P B119871119896+1(x) B

119871119896+1(y) 119886119903119890 (119864119898) -119878

lt1

4119871minus119875(119873119896+1)

119896+1

(119)

Proof Let

R(2)= exist119864 B

119871119896+1(x) B

119871119896+1(y) 119886119903119890 119864-R

S(2)= B

119871119896+1(x) B119871119896+1

(y) 119886119903119890 (119864119898) -S (120)

ThenPS(2) le PR(2)

+PS(2)R(2)


119871119896+1(x) B

119871119896+1(y) must be 119864-NR


B(119873)

119871119896+1(x) = B(119899

1015840)

119871119896+1(x1015840) times B(119899

1015840)

119871119896+1(x10158401015840) (121)


119871119896+1(x10158401015840) are119898-Loc and

B119871119896+1(x) is (119864119898)-S then B




B(119873)

119871119896+1(y) = B(119899

1015840)

119871119896+1(y1015840) times B(119899

10158401015840)

119871119896+1(y10158401015840) (122)

If both B(1198991015840)119871119896+1(y1015840) and B(11989910158401015840)

119871119896+1(y10158401015840) are 119898-Loc and

B(119873)

119871119896+1(y) is (119864119898)-S then B(119873)

119871119896+1(y)must be 119864-R Let


1015840)

119871119896+1(x1015840) B(119899

10158401015840)

119871119896+1(x10158401015840)

B(1198991015840)

119871119896+1(y1015840) B(119899

1015840)

119871119896+1(y10158401015840) is 119898-NL119900119888

(123)


P L(4) le 4119871

minus119875(119873minus1119896+1)

119896+1= 4119871

minus2119875(119873119896+1)

119896+1 (124)



large enough

P S(2) le P R

(2) + P L

(4)

le eminus119871120573

119896 + 4119871minus2119875(119873119896+1)

119896+1lt1

4119871minus119875(119873119896+1)

119896+1

(125)






B119871119896(k) sub B

119871119896+1(u)with 120588(w k) ge 119862

1198731198711+120575

119896is (119864119898)-NS In the







119896also large)

1198711+984858

119896≫ 119871

1+120575


120588 (xB(119862119873minus1)119871119896

(w)) + 120588 (yB(119862119873minus1)119871119896(w))

ge 119877 minus (2119862119873minus 2) 119871

1+120575

119896

(126)

Assume first that

1199031015840= 120588 (xB

(119862119873minus1)1198711+120575119896(w)) ge 119871

119896+ 1

11990310158401015840= 120588 (yB

(119862119873minus1)1198711+120575119896(w)) ge 119871119896 + 1

(127)


1199031015840(x) and in B

11990310158401015840(y) are


1)1198711+120575

119896minus 2 ge 119877 minus 2119862

1198731198711+120575

119896










(119877minus21198621198731198711+120575119896 minus2119871119896)(119871119896+1)

sdot e119871120573

119896+1] = 119898(1 +1

2119871minus120591

119896)

119869119871119896

119871119896+ 1119877(1 minus

3119862119873

119877

sdot 119871119896) minus 119871

120573

119896+1= 119898119877[(1 +

1

2119871minus120591

119896)

119869


119896)


119873119871119896

1198711+984858

119896

) minus119871120573

119896+1

119898119877] = 119898119877[(1 +

1

2119871minus120591

119896)

119869


119896) (1 minus 3119862

119873119871minus984858

119896) minus

119871120573

119896+1

1198981198711+984858

119896

] ge 119898(1 +1

4


119896)

119869

119877 ge 120574 (119898 119871119896+1) 119877

(128)






1 119909

119873) 119909





1003816100381610038161003816 le 119862119889119871119889 (129)





by the functions

Φa =1

radic119873

sum

120587isinS119873

119873

⨂

119895=1

1119886120587minus1(119895) (130)

where a = 1198861 119886

119873 with 119886

1 119886

119873 = 119873



set is

ZNgt

= a = 1198861 119886

119873 119886

119895isinZ 119886

1 119886

119873 = 119873

(131)



1 119888

2 119888

119873 b = 119887

1 119888

2 119888

119873 with

1198861 119887

1 119888

2 119888

119873 = 119873 + 1

(ii) dZ(1198861 1198871) = 1


120588 (x y) = min120587isinS119873

max1le119895le119873

dZ (119909120587minus1(119895) 119910119895) (132)




119892V(x 120596) + U(x) where

U (x) = sum119894 =119895

119880(d (119909119894 119909

119895)) 119880 N 997888rarr R (133)


V (x 120596) = 119881 (1199091 120596) + sdot sdot sdot + 119881 (119909

119873 120596) (134)






Appendix



V (1199091 119909

119873 120596) = sum

119910isinxn119910119881 (119910 120596) (A1)





119908(y)) 119881 (119908 120596)

+ sum

V =119908

119888V119881 (V 120596) (A2)


119888V = nV (x) minus nV (y) (A3)



119908 while

n119908(x) minus n

119908(y) = 119888

119908isin Z 0 hence 1003816100381610038161003816119888119908

1003816100381610038161003816 ge 1 (A4)

Therefore

P 1003816100381610038161003816119892V (x 120596) minus 119892V (y 120596)

1003816100381610038161003816 le 119904

= E [P 1003816100381610038161003816V (x 120596) minus V (y 120596)

1003816100381610038161003816 le10038161003816100381610038161198921003816100381610038161003816

minus1119904 | F

=119908]

= E [P 1003816100381610038161003816119888119908119881 (119908 120596) | z (120596)

1003816100381610038161003816 le10038161003816100381610038161198921003816100381610038161003816

minus1119904 | F

=119908]

(A5)




120596 119888119908119881 (119908 120596) isin [z (120596) minus

10038161003816100381610038161198921003816100381610038161003816

minus1119904 z (120596) +

10038161003816100381610038161198921003816100381610038161003816

minus1119904]

= 120596 119881 (119908 120596)

isin [z (120596) minus11990410038161003816100381610038161198921003816100381610038161003816

minus1

119888119908

z (120596) +11990410038161003816100381610038161198921003816100381610038161003816

minus1

119888119908

]

(A6)


P 1003816100381610038161003816V (x 120596) minus V (y 120596)

1003816100381610038161003816 le10038161003816100381610038161198921003816100381610038161003816

minus1119904

le sup119886isinR

(119865119881119908(119886 + 2119904) minus 119865

119881119908(119886))

(A7)


at 119908 given F=119908 Since 119865




1003816100381610038161003816 le 119904 (A8)


1198710(u) admits



0 To prove the




100381610038161003816100381610038161003816GB1198710 (u)

(x y 119864)100381610038161003816100381610038161003816le eminus119898(119904)d(xy)

(A9)





119899

119895=11198612119871(119909

119895) If




1198711+120575(x) can be


(⋃

119895isinJ

1198612119871(119909

119895))∐(⋃

119894isinJ119888

1198612119871(119909

119894)) (A10)


120588 (119861119871(119909

119894) 119861

119871(119909

119895)) gt 2119871 (A11)


119873119871 then for some 119895

∘ 120588(119909

119895∘ 119910

119895∘) gt 119862

119873119871


120588 (119909119895 119909

119895∘) le diamΠx le 119860

119873119871

120588 (119910119895 119910

119895∘) le diamΠy lt 119860

119873119871

(A12)


120588 (119909119894 119910

119895) ge 120588 (119909

119895∘ 119910

119895∘) minus 120588 (119909

119894 119909

119895∘) minus 120588 (119910

119895 119910

119895∘)

ge 119862119873119871 minus 2119860

119873119871 ge 3119871

(A13)


yielding

120588 (Π119861119871(x) Π119861

119871(y)) gt 3119871 minus 2119871 = 119871 (A14)



Acknowledgments


References

































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












Φa (x) =1

radic119873

sum

120587isinS119873

(minus1)120587

119873

⨂

119895=1

1119886120587minus1(119895) (4)


119873 = 119873 and 1




120587minus1(1) 119886



gt) of square-




119873


ge exist119894 = 119895 119909

119894=




119886119873 a isinZN

gt (5)


119873




(ΔG119891) (119909) = sum

⟨119909119910⟩

(119891 (119910) minus 119891 (119909))

= minus119899G (119909) 119891 (119909) + sum

⟨119909119910⟩

119891 (119910)

(6)



1= |119909

1minus 119910


119873minus 119910




Δ =

119873

sum

119895=1

(

119895minus1

⨂

119894=1

1(119894)) otimes Δ(119895) otimes (119873

⨂

119896=119895+1

1(119896)) (7)


(Δ(119895)119891) (119909

119895) = minus2119891 (119909

119895) + 119891 (119909

119895minus 1) + 119891 (119909

119895+ 1)

119909119895isin Z

(8)





119895minus 119887

119895| = 1 while for all 119894 = 119895 119886

119894= 119887

119894



restriction to ZNgt







defined by

120588 (x y) = max1le119895le119873

dZ (119909119895 119910119895) (9)


B119871(x) = y 120588 (x y) le 119871 =

119873

⨉

119895=1

119861119871(119909

119895) (10)

Here 119861119871(119909) = [119909 minus 119871 119909 + 119871] cap Z





119899


gtand x10158401015840 isinZ(n10158401015840)

gtin a usual way

dist (x1015840 x10158401015840) = min119906isinx1015840



x1015840 isin Z(n1015840)gt

and x10158401015840 isin Z(n10158401015840)gt


B(119873)

119871(x) = B(119899

1015840)

119871(x1015840) times B(119899

10158401015840)

119871(x10158401015840) (12)









gt(yΛ119888) = 1

120597+Λ = 120597

minusΛ

119888


(13)


1 119909


nx(119910) = 119895 119909119895= 119910 119910 isin X


gt letΣ(H


Λ

andGΛ(119864) = (H

Λminus119864)







H (120596) = H0+W (14)




119892V (x 120596) = 119892119881 (1199091 120596) + sdot sdot sdot + 119892119881 (119909

119873 120596) (16)



U (x) = sum119894 =119895

119880(10038161003816100381610038161003816119909119894minus 119909

119895

10038161003816100381610038161003816) (17)


0 lt infin




tions




119865119881119909(119905) = P 119881 (119909 120596) le 119905 119909 isinZ (18)


119865119881119909(119905 | F

=119909) = P 119881 (119909 120596) le 119905 | F

=119909 (19)





sup119909isinZ

sup119886isinR

[119865119881119909(119886 + 119904 | F

119881

=119909) minus 119865

119881119909(119886 | F

119881

=119909)] le 119862119904 (20)



Λthe



1015840 119861

10158401015840) ge 119871 ge 1 any events E1015840

isin

F119881

1198611015840 E10158401015840

isin F119881

11986110158401015840 and some 119862 gt 0

10038161003816100381610038161003816P E

1015840capE

10158401015840 minus P E

1015840P E

1015840101584010038161003816100381610038161003816le eminus119862 ln2119871

(21)



eminus119862 ln2(1198710)120572119896

lt ((1198710)120572119896

)

minus119886119901(1+120579)119896

119896 ge 0 (22)








120585119876(120596) = |119876|

minus1sum

119909isin119876

119881 (119909 120596) (23)


119876(120596)119909 isin 119876 Denote byF


generated by 120578119909 119881

119910 119909 isin 119876 119910 notin 119876


(W3) There exist1198621015840 119860

1015840 119887




P 120585119876(120596) isin [120582 (120596) 120582 (120596) + 119904] le 119862

1015840|119876|

1198601015840

1199041198871015840

(24)









1lowast 120583

2 where 120583

2is



0 lt 120588lowastle 120588

1 (119905) le 120588 lt infin

1205881015840

1(119905)

1205881(119905)le 119862

1lt infin

forall119905 isin (119886 119887)

(25)




119880 (119903) = 0 forall119903 ge 1199030 (26)


|119880 (119903)| le 119862eminus1198881199031minus120577

forall119903 ge 0 (27)


26 Main Results








119895 x)

large enough

10038161003816100381610038161003816Ψ

119895(x 120596)10038161003816100381610038161003816 le eminus119898120588(x119895x) 119898 gt 0 (28)


E[sup119905isinR

10038161003816100381610038161003816⟨1y10038161003816100381610038161003816eminusi119905H(120596)10038161003816100381610038161003816

1x⟩10038161003816100381610038161003816] le 119862eminus119886 ln

1+119888120588(xy) (29)


E[sup119905isinR

100381710038171003817100381710038171003817e(1198862) ln








H(120596) on 119868




119873

y = 1199101 119910

119873


dZ (119909119894 y) max119894

dZ (119910119894 x)] (31)

where

dZ (119909 y) = min119895

dZ (119909 119910119895) (32)


E[sup119905isinR

10038161003816100381610038161003816⟨1y10038161003816100381610038161003816eminusi119905H(120596)10038161003816100381610038161003816

1x⟩10038161003816100381610038161003816] le 119862 (x) eminus119886 ln

1+119888dH(xy) (33)



119871(119904) le 119862119871

119860119904119899 119862119860 isin

(0 +infin) and 119887 isin (0 1]





gt |Λ| equiv


HΛ (120596) = 1

ΛH (120596) 1Λ HΛ (34)







(Γxy119891) (x) = 120575xy119891 (y) (35)


Λ(119864) = (H

Λminus119864)




10038161003816100381610038161003816

le 119862ℓ


10038161003816100381610038161003816GBℓ(x) (x k)

10038161003816100381610038161003816max

k1015840isin120597+Bℓ(x)

10038161003816100381610038161003816GB119871 (k

1015840 y)10038161003816100381610038161003816

(36)

where

119862ℓ=1003816100381610038161003816120597Bℓ

(k)1003816100381610038161003816 le 119862 (119873) ℓ119873 (37)


10038161003816100381610038161003816G

119861119871(x y)10038161003816100381610038161003816

le (119862ℓmax

120588(xk)=ℓ

10038161003816100381610038161003816GBℓ(x) (x k)

10038161003816100381610038161003816) max

120588(uk1015840)leℓ+1

10038161003816100381610038161003816GB119871 (k

1015840 y)10038161003816100381610038161003816

(38)


10038161003816100381610038161003816GB119871 (x y 119864)

10038161003816100381610038161003816

le 119862ℓ

10038171003817100381710038171003817GBℓ(u) (119864)

10038171003817100381710038171003817max

k120588(uk)leℓ+1

10038161003816100381610038161003816GB119871 (k y 119864)

10038161003816100381610038161003816

(39)


ℓ(u)

1003816100381610038161003816120595 (x)1003816100381610038161003816 le 119862ℓ

10038171003817100381710038171003817GBℓ(x) (119864)

10038171003817100381710038171003817max

y120588(xy)leℓ+11003816100381610038161003816120595 (y)

1003816100381610038161003816 (40)



gt


119877(119906) rarr R



ℓ(119909) sub Λ

119877(119906) one has

119891 (119909) le 119902 max119910d(119909119910)leℓ

119891 (119910) (41)





M (119891 119861)

le 119902(119871minusℓ)(ℓ+1)

M (119891 119861)

M (119891 Λ) = max119909isinΛ

1003816100381610038161003816119891 (119909)1003816100381610038161003816

(42)





M (119891 119861)

(43)





1015840) ⊊ Λ and in 11990910158401015840 isin 119861

11990310158401015840(119906

10158401015840) ⊊ Λ with

d(1199061015840 11990610158401015840) ge 1199031015840 + 11990310158401015840 + 2 Then

119891 (1199061015840 119906

10158401015840) le 119902


10158401015840+1)(ℓ+1)rfloor

M (119891 119861)

le 119902(1199031015840+11990310158401015840minus2ℓ)(ℓ+1)

M (119891 119861)

(44)



tion

11989111991010158401015840 119910

1015840997891997888rarr 119891 (119910

1015840 119910

10158401015840) (45)


1015840) ⊊ Λ so by Lemma 6

11989111991010158401015840 (119906

1015840) = 119891 (119906

1015840 119910

10158401015840) le 119902

lfloor(1199031015840+1)(ℓ+1)rfloor

M (119891 Λ times Λ) (46)


10158401015840997891rarr 119891(119906

1015840 119910

10158401015840) which is

(ℓ 119902)-dominated in 11986111990310158401015840(119906






119896+1=

lceil119871120572

119896rceil 119871



0to be large enough


119871(u) is called




ℓ(u) sube B

119871(u) with ℓ ge 1198711120572





10038161003816100381610038161003816GB119871(u) (u y 119864 120596)

10038161003816100381610038161003816le eminus120574(119898119871)119871+2119871

120573

(47)

where

120574 (119898 119871) = 119898 (1 + 119871minus120591) 120591 =

1

8 (48)




120595119895 119895 = 1 |B



0isin B

119871119896(u)withd(x

0 y

0) gt 119871

119896minus1+1

and an integer

119877 isin [119871119896minus1 d (x

0 y

0) minus 119871

119896minus1+ 1] (49)


119877(x

0) is (119864119898)-NS and set

119902 = eminus120574(119898ℓ)ℓ (50)

Then



120595119895=

|120595119895⟩⟨120595





B119877(x

0) ni x 997891997888rarr

100381610038161003816100381610038161003816GB119871119896 (u)

(x y 119864)100381610038161003816100381610038161003816

(51)


0) and globally



119877(y


0fixed




satisfies10038161003816100381610038161003816120595

119895(x)120595

119895(y)10038161003816100381610038161003816 le eminus120574(119898119871)120588(xy) (52)




120588(x y) ge 119862119873119871 with 119862

119873= 2119860

119873+ 3 119860

119873= 4119873



in Section 43)



ℓ(y) and119898-







1003816100381610038161003816GB (x y 119864)1003816100381610038161003816 le sum

119864119895isinΣ(HB)

10038161003816100381610038161003816120595

119895(x)1003816100381610038161003816100381610038161003816100381610038161003816120595

119895(y)10038161003816100381610038161003816

10038161003816100381610038161003816119864 minus 119864

119895

10038161003816100381610038161003816

(53)


120573


1003816100381610038161003816GB (x y 119864)1003816100381610038161003816 le eminus120574(119898119871)119871+119871

120573+ln|B|

le eminus120574(119898119871)119871+2119871120573

(54)



119898(1 + 119871minus120591

119896) 119871

119896minus 2119871

120573

119896= 119898119871

119896+ 119898119871

1minus120591

119896minus 2119871

120573

119896

ge 119898(1 +1

2119871minus120591

119896) 119871

119896

(55)




(1)




(119864119898)-NS(B) if a ball B


(119864119898)-NS




119871119896(k) sub B

119871119896(u) with






120588 (xB(119862119873minus1)119871119896

(w)) + 120588 (yB(119862119873minus1)119871119896(w))

ge 119877 minus (2119862119873minus 2) 119871

119896

(56)

Assume first that

1199031015840= 120588 (xB

(119862119873minus1)119871119896(w)) ge 119871

119896+ 1

11990310158401015840= 120588 (yB

(119862119873minus1)119871119896(w)) ge 119871

119896+ 1

(57)



119896are auto-


Furthermore

1199031015840+ 119903

10158401015840ge 119877 minus 2 (119862

119873minus 1) 119871

119896minus 2 ge 119877 minus 2119862

119873119871119896 (58)










(119877minus2119862119873119871119896minus2119871119896)(119871119896+1)

e119871120573

119896+1]

= 119898(1 +1

2119871minus120591

119896)119871119896

119871119896+ 1119877 (1 minus 3C

119873119877minus1119871119896) minus 119871

120573

119896+1

= 119898119877[(1 +1

2119871minus120591

119896) (1 minus 119871

minus1

119896) (1 minus 3119862

119873119871minus984858

119896)

minus119871120573

119896+1

119898119877] ge 119898(1 +

1

4119871minus120591

119896)119877 ge 120574 (119898 119871

119896+1) 119877

(59)




B119871119896(x) B


119871119896+1(u) is 119864-


119871119896(y) must


the hypothesis






119895 (k1015840 k10158401015840) 997891rarr |120595

119895(k1015840)120595

119895(k10158401015840)| where

120595119895 119895 isin [1 |B





119895 = 1 so the


119871119896+1(u) and

assume 120588(x y) = 119877 ge 1198711+984858119896




minus ln 10038161003816100381610038161003816120595119895 (x)120595119895 (y)10038161003816100381610038161003816


(119877minus2119862119873119871119896minus2119871119896)(119871119896+1)

]

(60)






120588 (x y) ge 1198711+984858119896minus1

0 lt 120591 lt 120588

1 + 984858 lt 120572

120573 lt 1 minus 120591

(61)


120572 =4

3

120588 =1

6

120591 =1

8

120573 =1

2

(62)




119901 gt 0 there is 1198920= 119892

0(119898 119901 119871


|119892| ge 1198920 any ball B

1198710(119906) and any 119864 isin R

P B1198710(u) 119894119904 (119864119898) -119878 le 119871minus119901

0

P B1198710(u) 119894119904 119898-NL119900119888 le 119871

minus119901

0

(63)




1015840isin (0 1] 1198601015840


119871(u1015840) B(119873)



P dist (Σ119871u1015840 Σ119871u10158401015840) le 119904 le (3119871)

2119873ℎ119871 (2119904) (64)


HB119871(u10158401015840) and

ℎ119871 (119904) = 119862

1015840101584011987111986010158401015840

119904119887and1198871015840

119887 and 1198871015840equiv min [119887 1198871015840] (65)

and 11986010158401015840= 119860

10158401015840(119860



u10158401015840) gt 4119873119871

P exist119864 isin R B119871(u1015840) B

119871(u10158401015840) 119886119903119890 119864-119875119877 le eminus119871

1205731015840

(66)





119873= 4119873 and





H(119873)

B119871(u) = H(1198991015840)

B119871(u1015840)otimes 1(119899

10158401015840)+ 1(119899

1015840)otimesH(119899

10158401015840)

B119871(u10158401015840)(67)

with 1198991015840+11989910158401015840 = 119873 u = (u1015840 u10158401015840) u1015840isinZn1015840

gt u10158401015840isinZn10158401015840

gt while for




(x) ΠJ119888B119871(x)) gt 2119871

(B) If B119871(x) B


then 120588(ΠB119871(x) ΠB

119871(y)) ge 2119871






B(119873)

119871119896(u) = B(119899

1015840)

119871119896(u1015840) times B(119899

10158401015840)

119871119896(u10158401015840) (68)


If the balls B(1198991015840)

119871119896minus1(u1015840) and B(119899

10158401015840)


B(119873)

119871119896(u) is also119898-Loc



HB119871119896 (u)= HB(119899

1015840)119871119896

(u1015840) otimes 1(11989910158401015840)+ 1(119899

1015840)otimesHB(119899

10158401015840)119871119896

(u10158401015840) (69)



119895= 120601

1198951015840 otimes120595

11989510158401015840 where 120601


1015840)119871119896



10158401015840)119871119896

(u10158401015840) Now


119895 (x)Ψ119895 (y)10038161003816100381610038161003816

=100381610038161003816100381610038161206011198951015840 (x1015840)120601

11989510158401015840 (x10158401015840)120595

1198951015840 (y1015840)120595

11989510158401015840 (y10158401015840)10038161003816100381610038161003816

(70)


1015840)

119871119896(u1015840) and B(119899

10158401015840)

119871119896(u10158401015840)


119875 (119899 119896) = 2119873minus119899119901 (1 + 120579)

119896 1 le 119899 le 119873 119896 ge 0 (71)



119871119896(u)

P B(119899)

119871119896(u) 119894119904 119898-NL119900119888 le 119871

minus119875(119899119896)

119896 (72)


119871119896(u) one has

P B(119873)

119871119896(u) 119894119904 119898-NL119900119888 le 2119871

minus2119875(119873119896)

119896le1

4119871minus119875(119873119896)

119896 (73)




1015840)

119871119896(u1015840) B(119899

10158401015840)

119871119896(u10158401015840) is 119898-NLoc


P B(119873)

119871119896(u) is 119898-NL119900119888

le P B(11989910158401015840)

119871119896(u10158401015840) is 119898-NL119900119888

+ P B(11989910158401015840)

119871119896(u10158401015840) is 119898-NL119900119888

le 119871minus119875(1198991015840119896)

119896+ 119871

minus119875(11989910158401015840119896)

119896le 2119871

minus2119875(119873119896)

119896lt1

4119871minus119875(119873119896)

119896

(74)





1198961015840 with 1198961015840 le 119896


119871119896(u)

P B(119873)

119871 119894(u) 119894119904 119898-NL119900119888 le 119871

minus119875(119873119894)

119894 (75)


119871119896+1(x) one has

P B(119873)

119871119896+1(u) 119894119904 119898-NL119900119888 le 119871

minus119875(119873119896+1)

119896+1 (76)

provided that

119901 gt2120572

2

2 minus 1205722119873119889

0 lt 3120579 le min2 minus 1205722

1205722minus2119873119889


(77)


119871119896(u) B(119873)

119871119896(k) one

has for some 119886 119888 gt 0


119871119896(u) B(119873)

119871119896(k) 119886119903119890 (119864119898) -119878

le 119871minus119875(119873119896)

119896equiv 119871

minus119901(1+120599)119896


1+119888119871119896

(78)


N119896+1= B(119873)

119871119896+1(u) is 119898-NLo119888

S(2)

119896


119871119896(u) B(119873)

119871119896(k)

sub B(119873)

119871119896+1(w)

R(2)


119871119896(u) B(119873)

119871119896(k)

sub B(119873)

119871119896+1(w)

(79)


N119896+1sub S

(2)

119896subR

(2)

119896cup (S

(2)

119896R

(2)

119896) (80)


119896 le eminus119871

1205731015840


119896R

(2)


119871119896+1(w) and


S(2)

119896(u k) = exist119864 B(119873)

119871119896(u) B(119873)

119871119896(k) are (119864119898) -S (81)


119896(u k) R(2)

119896 either B(119873)

119871119896(u) or B(119873)

119871119896(k)


119871119896(u) is 119864-CNR



pair of 119898-NL119900119888 balls B(119873)

119871119896minus1(y1015840) B(119873)


(12)1198711+120575


(1) Both B(119873)

119871119896minus1(y1015840) and B(119873)

119871119896minus1(y10158401015840) are FI


120588 (ΠB119871119896minus1(u) ΠB

119871119896minus1(k)) ge 2119871

119896 (82)


119896minus1= 119871

11205722

119896+1and 119871

0large enough

P B(119873)

119871119896minus1(y1015840) B(119873)

119871119896minus1(y10158401015840) are 119898-NL119900119888

le 2119871minus(2119901120572

2)(1+120579)

119896minus1

119896+1

(83)


119871119896(w)|2 le (321198732)1198712119873


P S(2)

119896R

(2)

119896 le32119873

2119871minus(2119901120572

2)(1+120579)

119896minus1+2119873

119896+1 (84)



119896+1 under con-


2119901

12057222119873minus119873119901 (1 + 120579)

119896minus1minus 2119873119889 gt 2

119873minus119873119901 (1 + 120579)

119896+1 (85)

or equivalently

(1 + 120579)2lt2119901

1205722minus

2119873119889

119901 (1 + 120579)119896minus1 (86)


3120599 le2 minus 120572

2

1205722minus2119873

119901 (87)


P S(2)

119896 le P R

(2)

119896 + P S

(2)

119896R

(2)

119896

le eminus1198711205731015840

119896 +1

4119871minus119875(119873119896+1)

119896+1lt1

2119871minus119875(119873119896+1)

119896+1

(88)


119871119896minus1(y1015840) orB(119873)


P B(119873)

119871119896minus1(y1015840) B(119873)

119871119896minus1(y10158401015840) is 119898-NL119900119888

le min (P B(119873)

119871119896minus1(y1015840) is 119898-NL119900119888

P B(119873)

119871119896minus1(y10158401015840) is 119898-NL119900119888) le 119871

minus2119901(1+120579)119896minus1

119896minus1

lt 119871minus119901(1+120579)

119896+1

119896+1

(89)



119871minus119875(119873119896)

119896equiv 119871

minus119901(1+120599)119896


1+119888119871119896 (90)




P B(119873)

119871119896(u) 119894119904 119898-NL119900119888 le

1

4119871minus119901(1+120579)

119896

119896(91)


minus119901(1+120579)119896


119895of operator HB(119873)119871119896 (u)



119896one

has10038161003816100381610038161003816Ψ119895(x) Ψ






1003816100381610038161003816Ψ (y)1003816100381610038161003816 le eminus119898y

(93)








119871119896(y) sub

B119871119896+2(z) Fix 120596 isin Ω1015840


2le 1 implies Ψ

infinle 1 there is


119899isin B

119871119896minus1(0) then






119871119896+2(z) with 120588(y z) isin [119871

119896+1 119871

119896+2)


1+120575

119896) is (120582119898)-NS Note that



119877(y) with

119877 = 120588(y z)minus21198711+120575119896minus1 gt (12)119871

1+2120575



minusln 1003816100381610038161003816Ψ (y)

1003816100381610038161003816

120588 (y z)ge 119898 (1 + 119871

minus120591

119896) (1 minus

2119871119896+ 1

120588 (y z))

ge 119898(1 +1

2119871minus120591

119896) gt 119898

(95)









119871(119909) B


P exist 119864 isin 119868 B119871(x) B

119871(y) 119886119903119890 (119864119898) -S le 119891 (119871) (96)


119871(119909) cup B

119871(119910) one has

E[ sup120601isinB1(119868)

10038161003816100381610038161003816⟨1x1003816100381610038161003816120601 (HΛ (120596))

1003816100381610038161003816 1y⟩10038161003816100381610038161003816]

le 119862119871119889119890minus119898119871+ 119891 (119871)

(97)




Λ sup B119871(119909) cup B

119871(119910)


10038161003816100381610038161003816⟨1x1003816100381610038161003816120601 (HΛ (120596))

1003816100381610038161003816 1y⟩10038161003816100381610038161003816] le 119862119890

minus119886 ln1+119888120588(xy) (98)










same 119888 119886 gt 0 as in (98)

E[ sup120601isinC1(119868)

10038161003816100381610038161003816⟨1x1003816100381610038161003816120601 (H (120596))

1003816100381610038161003816 1y⟩10038161003816100381610038161003816] le 119862eminus119886 ln

1+119888120588(xy) (99)




int120601 (120582) 119889120583xyB120596 (120582)

= ⟨1x1003816100381610038161003816 120601 (HB (120596))Π119868

(HB (120596))100381610038161003816100381610038161y⟩

(100)








119899rarr H with







119871119896

E [1003816100381610038161003816120583

xy120596


E [100381610038161003816100381610038161003816120583xyB119871119896

100381610038161003816100381610038161003816(E)] (101)




120601119905 120582 997891997888rarr 119891

119868(120582) ei119905120582 119905 isin R (102)








984858 = 2120575

120572 = 1 + 4120575

120591 =120575

2

(103)



120574 (119898 119871 119899) = 119898 (1 + 119871minus120591)119873minus119899+1

(104)



120588(x y) ge 1198621198731198711+120575 with 119862

119873= 4119873






119871(x) admits

a factorization

B(119873)

119871(x) = B(119899

1015840)

119871(x1015840) times B(119899

10158401015840)

119871(x10158401015840) (105)


119871(x) B


then 120588(ΠB119871(x) ΠB

119871(y)) ge 1198711+120575




119877119877 ge 119903



119880(2)

119877(119903) = 1

119903le119877119880

(2)(119903) (106)


gt



119877(x1015840) + U



11987110158401015840(x10158401015840) with

120588(B1198711015840(x1015840)B



H(119877)

B = H(119877)

B1198711015840(x1015840) otimes 1

(11989910158401015840)+ 1(119899

1015840)otimesH(119877)

B11987110158401015840

(x10158401015840) (107)


B(119873)


119873= 4119898119873


119871(x) =

B1198711015840(x1015840) times B

11987110158401015840(x10158401015840) with dist(x1015840 x10158401015840) gt (119862

1198732119873)119871 = 2119898119871

so that with 119877119896= (119862

1198732119873)119871

119896 one has

120598 (119877119896) lt eminus2119898119871119896 lt

1

2eminus119871120573

119896 (108)


119871119896(u) =

B(1198991015840)

119871119896(uJ)timesB

(11989910158401015840)



(a) dist(x1015840 x10158401015840) gt 119877119896= (119862

1198732119873)119871

119896

(b) B(1198991015840)

119871119896(u1015840) and B(119899

10158401015840)

119871119896(u10158401015840) are119898-Loc

(c) B(119873)

119871119896(u) is 119864-NR


119871119896(u) is (119864119898)-NS

Proof Set 119877119896= (119862

1198732119873)119871


above Denote

H(119877119896119873)

u119896 = H(119877119896)

B(119873)119871119896 (u)

G (119864) = (HB(119873)119871119896 (u)minus 119864)

minus1

(109)

and G(119877119896)(119864) = (H(119877119896119873)



H(119877119896 119873)

u119896 = HB(1198991015840)119871119896

(u1015840) otimes 1(11989910158401015840)+ 1(119899

1015840)otimesHB(119899

1015840)119871119896

(x10158401015840) (110)

with u = (u1015840 u10158401015840) isin Z119889119899

1015840

times Z11988911989910158401015840


119886119887= 120582

119886+120583

119887 where 120582

119886 = Σ(HB119871119896 (u



119887 = Σ(HB119871119896 (u


of H(119877119896 119873)


119886otimes 120595

119887 where


1015840)119871119896


of HB(11989910158401015840)119871119896


119886+ 120583

119887




|119864 minus (120582119886+ 120583

119887)| ge e119871

120573


119887)| ge e119871

120573

119896 and alsoas |(119864 minus 120583

119887) minus 120582

119886)| ge e119871

120573


G(119877119896) (u y 119864)

= sum

120582119886

sum

120583119887

120601119886(u1015840)120601

119886(y1015840)120595

119887(u10158401015840)120595

119887(y10158401015840)

(120582119886+ 120583

119887) minus 119864

= sum

120582119886

P1015840

119886(u1015840 y1015840)G(119877119896)

B119871119896 (u10158401015840)(u10158401015840 y10158401015840 119864 minus 120582

119886)

= sum

120583119887

P10158401015840

119887(u10158401015840 y10158401015840)G(119877119896)

B119871119896 (u1015840)(u1015840 y1015840 119864 minus 120583

119887)

(111)


B119871119896 (u1015840)(119864minus120583

119887) andG(119877119896)

B119871119896 (u10158401015840)(119864minus120582

119886) are

nonresonant1003817100381710038171003817100381710038171003817G(119877119896)

B119871119896 (u1015840)(119864 minus 120583

119887)

1003817100381710038171003817100381710038171003817le e119871

120573

119896

1003817100381710038171003817100381710038171003817G(119877119896)

B119871119896 (u10158401015840)(119864 minus 120582

119886)

1003817100381710038171003817100381710038171003817le e119871

120573

119896

(112)


y1015840) = 119871119896 in which case we

infer from (b)

10038161003816100381610038161003816G(119877119896) (u y 119864)10038161003816100381610038161003816 le

10038161003816100381610038161003816B119871119896(u10158401015840)10038161003816100381610038161003816eminus120574(119898119871119896119873minus1)119871119896+119871

120573

119896 (113)

or 120588(u10158401015840 y10158401015840) = 119871


10038161003816100381610038161003816G(119877119896) (u y 119864)10038161003816100381610038161003816 le

10038161003816100381610038161003816B119871119896(u1015840)10038161003816100381610038161003816eminus120574(119898119871119896119873minus1)119871119896+119871

120573

119896 (114)



119871119896+ 119871

120573

119896+ 119862 ln 119871

119896)

lt1

2eminus120574(119898119871119896119873)

(115)




10038171003817100381710038171003817G minus G(119877119896)

10038171003817100381710038171003817le10038171003817100381710038171003817U minus U(119877119896)

10038171003817100381710038171003817

10038171003817100381710038171003817G(119877119896)

10038171003817100381710038171003817G le eminus2119898119871119896e2119871

120573

119896

le1

2eminus120574(119898119871119896119873)

(117)


10038161003816100381610038161003816G (x y 119864) minus G(119877119896) (x y 119864)10038161003816100381610038161003816 le

1

2eminus120574(119898119871119896 119873)


119871119896+1(u)

(118)

Lemma 33 Let B119871119896+1(x)B



P B119871119896+1(x) B

119871119896+1(y) 119886119903119890 (119864119898) -119878

lt1

4119871minus119875(119873119896+1)

119896+1

(119)

Proof Let

R(2)= exist119864 B

119871119896+1(x) B

119871119896+1(y) 119886119903119890 119864-R

S(2)= B

119871119896+1(x) B119871119896+1

(y) 119886119903119890 (119864119898) -S (120)

ThenPS(2) le PR(2)

+PS(2)R(2)


119871119896+1(x) B

119871119896+1(y) must be 119864-NR


B(119873)

119871119896+1(x) = B(119899

1015840)

119871119896+1(x1015840) times B(119899

1015840)

119871119896+1(x10158401015840) (121)


119871119896+1(x10158401015840) are119898-Loc and

B119871119896+1(x) is (119864119898)-S then B




B(119873)

119871119896+1(y) = B(119899

1015840)

119871119896+1(y1015840) times B(119899

10158401015840)

119871119896+1(y10158401015840) (122)

If both B(1198991015840)119871119896+1(y1015840) and B(11989910158401015840)

119871119896+1(y10158401015840) are 119898-Loc and

B(119873)

119871119896+1(y) is (119864119898)-S then B(119873)

119871119896+1(y)must be 119864-R Let


1015840)

119871119896+1(x1015840) B(119899

10158401015840)

119871119896+1(x10158401015840)

B(1198991015840)

119871119896+1(y1015840) B(119899

1015840)

119871119896+1(y10158401015840) is 119898-NL119900119888

(123)


P L(4) le 4119871

minus119875(119873minus1119896+1)

119896+1= 4119871

minus2119875(119873119896+1)

119896+1 (124)



large enough

P S(2) le P R

(2) + P L

(4)

le eminus119871120573

119896 + 4119871minus2119875(119873119896+1)

119896+1lt1

4119871minus119875(119873119896+1)

119896+1

(125)






B119871119896(k) sub B

119871119896+1(u)with 120588(w k) ge 119862

1198731198711+120575

119896is (119864119898)-NS In the







119896also large)

1198711+984858

119896≫ 119871

1+120575


120588 (xB(119862119873minus1)119871119896

(w)) + 120588 (yB(119862119873minus1)119871119896(w))

ge 119877 minus (2119862119873minus 2) 119871

1+120575

119896

(126)

Assume first that

1199031015840= 120588 (xB

(119862119873minus1)1198711+120575119896(w)) ge 119871

119896+ 1

11990310158401015840= 120588 (yB

(119862119873minus1)1198711+120575119896(w)) ge 119871119896 + 1

(127)


1199031015840(x) and in B

11990310158401015840(y) are


1)1198711+120575

119896minus 2 ge 119877 minus 2119862

1198731198711+120575

119896










(119877minus21198621198731198711+120575119896 minus2119871119896)(119871119896+1)

sdot e119871120573

119896+1] = 119898(1 +1

2119871minus120591

119896)

119869119871119896

119871119896+ 1119877(1 minus

3119862119873

119877

sdot 119871119896) minus 119871

120573

119896+1= 119898119877[(1 +

1

2119871minus120591

119896)

119869


119896)


119873119871119896

1198711+984858

119896

) minus119871120573

119896+1

119898119877] = 119898119877[(1 +

1

2119871minus120591

119896)

119869


119896) (1 minus 3119862

119873119871minus984858

119896) minus

119871120573

119896+1

1198981198711+984858

119896

] ge 119898(1 +1

4


119896)

119869

119877 ge 120574 (119898 119871119896+1) 119877

(128)






1 119909

119873) 119909





1003816100381610038161003816 le 119862119889119871119889 (129)





by the functions

Φa =1

radic119873

sum

120587isinS119873

119873

⨂

119895=1

1119886120587minus1(119895) (130)

where a = 1198861 119886

119873 with 119886

1 119886

119873 = 119873



set is

ZNgt

= a = 1198861 119886

119873 119886

119895isinZ 119886

1 119886

119873 = 119873

(131)



1 119888

2 119888

119873 b = 119887

1 119888

2 119888

119873 with

1198861 119887

1 119888

2 119888

119873 = 119873 + 1

(ii) dZ(1198861 1198871) = 1


120588 (x y) = min120587isinS119873

max1le119895le119873

dZ (119909120587minus1(119895) 119910119895) (132)




119892V(x 120596) + U(x) where

U (x) = sum119894 =119895

119880(d (119909119894 119909

119895)) 119880 N 997888rarr R (133)


V (x 120596) = 119881 (1199091 120596) + sdot sdot sdot + 119881 (119909

119873 120596) (134)






Appendix



V (1199091 119909

119873 120596) = sum

119910isinxn119910119881 (119910 120596) (A1)





119908(y)) 119881 (119908 120596)

+ sum

V =119908

119888V119881 (V 120596) (A2)


119888V = nV (x) minus nV (y) (A3)



119908 while

n119908(x) minus n

119908(y) = 119888

119908isin Z 0 hence 1003816100381610038161003816119888119908

1003816100381610038161003816 ge 1 (A4)

Therefore

P 1003816100381610038161003816119892V (x 120596) minus 119892V (y 120596)

1003816100381610038161003816 le 119904

= E [P 1003816100381610038161003816V (x 120596) minus V (y 120596)

1003816100381610038161003816 le10038161003816100381610038161198921003816100381610038161003816

minus1119904 | F

=119908]

= E [P 1003816100381610038161003816119888119908119881 (119908 120596) | z (120596)

1003816100381610038161003816 le10038161003816100381610038161198921003816100381610038161003816

minus1119904 | F

=119908]

(A5)




120596 119888119908119881 (119908 120596) isin [z (120596) minus

10038161003816100381610038161198921003816100381610038161003816

minus1119904 z (120596) +

10038161003816100381610038161198921003816100381610038161003816

minus1119904]

= 120596 119881 (119908 120596)

isin [z (120596) minus11990410038161003816100381610038161198921003816100381610038161003816

minus1

119888119908

z (120596) +11990410038161003816100381610038161198921003816100381610038161003816

minus1

119888119908

]

(A6)


P 1003816100381610038161003816V (x 120596) minus V (y 120596)

1003816100381610038161003816 le10038161003816100381610038161198921003816100381610038161003816

minus1119904

le sup119886isinR

(119865119881119908(119886 + 2119904) minus 119865

119881119908(119886))

(A7)


at 119908 given F=119908 Since 119865




1003816100381610038161003816 le 119904 (A8)


1198710(u) admits



0 To prove the




100381610038161003816100381610038161003816GB1198710 (u)

(x y 119864)100381610038161003816100381610038161003816le eminus119898(119904)d(xy)

(A9)





119899

119895=11198612119871(119909

119895) If




1198711+120575(x) can be


(⋃

119895isinJ

1198612119871(119909

119895))∐(⋃

119894isinJ119888

1198612119871(119909

119894)) (A10)


120588 (119861119871(119909

119894) 119861

119871(119909

119895)) gt 2119871 (A11)


119873119871 then for some 119895

∘ 120588(119909

119895∘ 119910

119895∘) gt 119862

119873119871


120588 (119909119895 119909

119895∘) le diamΠx le 119860

119873119871

120588 (119910119895 119910

119895∘) le diamΠy lt 119860

119873119871

(A12)


120588 (119909119894 119910

119895) ge 120588 (119909

119895∘ 119910

119895∘) minus 120588 (119909

119894 119909

119895∘) minus 120588 (119910

119895 119910

119895∘)

ge 119862119873119871 minus 2119860

119873119871 ge 3119871

(A13)


yielding

120588 (Π119861119871(x) Π119861

119871(y)) gt 3119871 minus 2119871 = 119871 (A14)



Acknowledgments


References

































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












gt(yΛ119888) = 1

120597+Λ = 120597

minusΛ

119888


(13)


1 119909


nx(119910) = 119895 119909119895= 119910 119910 isin X


gt letΣ(H


Λ

andGΛ(119864) = (H

Λminus119864)







H (120596) = H0+W (14)




119892V (x 120596) = 119892119881 (1199091 120596) + sdot sdot sdot + 119892119881 (119909

119873 120596) (16)



U (x) = sum119894 =119895

119880(10038161003816100381610038161003816119909119894minus 119909

119895

10038161003816100381610038161003816) (17)


0 lt infin




tions




119865119881119909(119905) = P 119881 (119909 120596) le 119905 119909 isinZ (18)


119865119881119909(119905 | F

=119909) = P 119881 (119909 120596) le 119905 | F

=119909 (19)





sup119909isinZ

sup119886isinR

[119865119881119909(119886 + 119904 | F

119881

=119909) minus 119865

119881119909(119886 | F

119881

=119909)] le 119862119904 (20)



Λthe



1015840 119861

10158401015840) ge 119871 ge 1 any events E1015840

isin

F119881

1198611015840 E10158401015840

isin F119881

11986110158401015840 and some 119862 gt 0

10038161003816100381610038161003816P E

1015840capE

10158401015840 minus P E

1015840P E

1015840101584010038161003816100381610038161003816le eminus119862 ln2119871

(21)



eminus119862 ln2(1198710)120572119896

lt ((1198710)120572119896

)

minus119886119901(1+120579)119896

119896 ge 0 (22)








120585119876(120596) = |119876|

minus1sum

119909isin119876

119881 (119909 120596) (23)


119876(120596)119909 isin 119876 Denote byF


generated by 120578119909 119881

119910 119909 isin 119876 119910 notin 119876


(W3) There exist1198621015840 119860

1015840 119887




P 120585119876(120596) isin [120582 (120596) 120582 (120596) + 119904] le 119862

1015840|119876|

1198601015840

1199041198871015840

(24)









1lowast 120583

2 where 120583

2is



0 lt 120588lowastle 120588

1 (119905) le 120588 lt infin

1205881015840

1(119905)

1205881(119905)le 119862

1lt infin

forall119905 isin (119886 119887)

(25)




119880 (119903) = 0 forall119903 ge 1199030 (26)


|119880 (119903)| le 119862eminus1198881199031minus120577

forall119903 ge 0 (27)


26 Main Results








119895 x)

large enough

10038161003816100381610038161003816Ψ

119895(x 120596)10038161003816100381610038161003816 le eminus119898120588(x119895x) 119898 gt 0 (28)


E[sup119905isinR

10038161003816100381610038161003816⟨1y10038161003816100381610038161003816eminusi119905H(120596)10038161003816100381610038161003816

1x⟩10038161003816100381610038161003816] le 119862eminus119886 ln

1+119888120588(xy) (29)


E[sup119905isinR

100381710038171003817100381710038171003817e(1198862) ln








H(120596) on 119868




119873

y = 1199101 119910

119873


dZ (119909119894 y) max119894

dZ (119910119894 x)] (31)

where

dZ (119909 y) = min119895

dZ (119909 119910119895) (32)


E[sup119905isinR

10038161003816100381610038161003816⟨1y10038161003816100381610038161003816eminusi119905H(120596)10038161003816100381610038161003816

1x⟩10038161003816100381610038161003816] le 119862 (x) eminus119886 ln

1+119888dH(xy) (33)



119871(119904) le 119862119871

119860119904119899 119862119860 isin

(0 +infin) and 119887 isin (0 1]





gt |Λ| equiv


HΛ (120596) = 1

ΛH (120596) 1Λ HΛ (34)







(Γxy119891) (x) = 120575xy119891 (y) (35)


Λ(119864) = (H

Λminus119864)




10038161003816100381610038161003816

le 119862ℓ


10038161003816100381610038161003816GBℓ(x) (x k)

10038161003816100381610038161003816max

k1015840isin120597+Bℓ(x)

10038161003816100381610038161003816GB119871 (k

1015840 y)10038161003816100381610038161003816

(36)

where

119862ℓ=1003816100381610038161003816120597Bℓ

(k)1003816100381610038161003816 le 119862 (119873) ℓ119873 (37)


10038161003816100381610038161003816G

119861119871(x y)10038161003816100381610038161003816

le (119862ℓmax

120588(xk)=ℓ

10038161003816100381610038161003816GBℓ(x) (x k)

10038161003816100381610038161003816) max

120588(uk1015840)leℓ+1

10038161003816100381610038161003816GB119871 (k

1015840 y)10038161003816100381610038161003816

(38)


10038161003816100381610038161003816GB119871 (x y 119864)

10038161003816100381610038161003816

le 119862ℓ

10038171003817100381710038171003817GBℓ(u) (119864)

10038171003817100381710038171003817max

k120588(uk)leℓ+1

10038161003816100381610038161003816GB119871 (k y 119864)

10038161003816100381610038161003816

(39)


ℓ(u)

1003816100381610038161003816120595 (x)1003816100381610038161003816 le 119862ℓ

10038171003817100381710038171003817GBℓ(x) (119864)

10038171003817100381710038171003817max

y120588(xy)leℓ+11003816100381610038161003816120595 (y)

1003816100381610038161003816 (40)



gt


119877(119906) rarr R



ℓ(119909) sub Λ

119877(119906) one has

119891 (119909) le 119902 max119910d(119909119910)leℓ

119891 (119910) (41)





M (119891 119861)

le 119902(119871minusℓ)(ℓ+1)

M (119891 119861)

M (119891 Λ) = max119909isinΛ

1003816100381610038161003816119891 (119909)1003816100381610038161003816

(42)





M (119891 119861)

(43)





1015840) ⊊ Λ and in 11990910158401015840 isin 119861

11990310158401015840(119906

10158401015840) ⊊ Λ with

d(1199061015840 11990610158401015840) ge 1199031015840 + 11990310158401015840 + 2 Then

119891 (1199061015840 119906

10158401015840) le 119902


10158401015840+1)(ℓ+1)rfloor

M (119891 119861)

le 119902(1199031015840+11990310158401015840minus2ℓ)(ℓ+1)

M (119891 119861)

(44)



tion

11989111991010158401015840 119910

1015840997891997888rarr 119891 (119910

1015840 119910

10158401015840) (45)


1015840) ⊊ Λ so by Lemma 6

11989111991010158401015840 (119906

1015840) = 119891 (119906

1015840 119910

10158401015840) le 119902

lfloor(1199031015840+1)(ℓ+1)rfloor

M (119891 Λ times Λ) (46)


10158401015840997891rarr 119891(119906

1015840 119910

10158401015840) which is

(ℓ 119902)-dominated in 11986111990310158401015840(119906






119896+1=

lceil119871120572

119896rceil 119871



0to be large enough


119871(u) is called




ℓ(u) sube B

119871(u) with ℓ ge 1198711120572





10038161003816100381610038161003816GB119871(u) (u y 119864 120596)

10038161003816100381610038161003816le eminus120574(119898119871)119871+2119871

120573

(47)

where

120574 (119898 119871) = 119898 (1 + 119871minus120591) 120591 =

1

8 (48)




120595119895 119895 = 1 |B



0isin B

119871119896(u)withd(x

0 y

0) gt 119871

119896minus1+1

and an integer

119877 isin [119871119896minus1 d (x

0 y

0) minus 119871

119896minus1+ 1] (49)


119877(x

0) is (119864119898)-NS and set

119902 = eminus120574(119898ℓ)ℓ (50)

Then



120595119895=

|120595119895⟩⟨120595





B119877(x

0) ni x 997891997888rarr

100381610038161003816100381610038161003816GB119871119896 (u)

(x y 119864)100381610038161003816100381610038161003816

(51)


0) and globally



119877(y


0fixed




satisfies10038161003816100381610038161003816120595

119895(x)120595

119895(y)10038161003816100381610038161003816 le eminus120574(119898119871)120588(xy) (52)




120588(x y) ge 119862119873119871 with 119862

119873= 2119860

119873+ 3 119860

119873= 4119873



in Section 43)



ℓ(y) and119898-







1003816100381610038161003816GB (x y 119864)1003816100381610038161003816 le sum

119864119895isinΣ(HB)

10038161003816100381610038161003816120595

119895(x)1003816100381610038161003816100381610038161003816100381610038161003816120595

119895(y)10038161003816100381610038161003816

10038161003816100381610038161003816119864 minus 119864

119895

10038161003816100381610038161003816

(53)


120573


1003816100381610038161003816GB (x y 119864)1003816100381610038161003816 le eminus120574(119898119871)119871+119871

120573+ln|B|

le eminus120574(119898119871)119871+2119871120573

(54)



119898(1 + 119871minus120591

119896) 119871

119896minus 2119871

120573

119896= 119898119871

119896+ 119898119871

1minus120591

119896minus 2119871

120573

119896

ge 119898(1 +1

2119871minus120591

119896) 119871

119896

(55)




(1)




(119864119898)-NS(B) if a ball B


(119864119898)-NS




119871119896(k) sub B

119871119896(u) with






120588 (xB(119862119873minus1)119871119896

(w)) + 120588 (yB(119862119873minus1)119871119896(w))

ge 119877 minus (2119862119873minus 2) 119871

119896

(56)

Assume first that

1199031015840= 120588 (xB

(119862119873minus1)119871119896(w)) ge 119871

119896+ 1

11990310158401015840= 120588 (yB

(119862119873minus1)119871119896(w)) ge 119871

119896+ 1

(57)



119896are auto-


Furthermore

1199031015840+ 119903

10158401015840ge 119877 minus 2 (119862

119873minus 1) 119871

119896minus 2 ge 119877 minus 2119862

119873119871119896 (58)










(119877minus2119862119873119871119896minus2119871119896)(119871119896+1)

e119871120573

119896+1]

= 119898(1 +1

2119871minus120591

119896)119871119896

119871119896+ 1119877 (1 minus 3C

119873119877minus1119871119896) minus 119871

120573

119896+1

= 119898119877[(1 +1

2119871minus120591

119896) (1 minus 119871

minus1

119896) (1 minus 3119862

119873119871minus984858

119896)

minus119871120573

119896+1

119898119877] ge 119898(1 +

1

4119871minus120591

119896)119877 ge 120574 (119898 119871

119896+1) 119877

(59)




B119871119896(x) B


119871119896+1(u) is 119864-


119871119896(y) must


the hypothesis






119895 (k1015840 k10158401015840) 997891rarr |120595

119895(k1015840)120595

119895(k10158401015840)| where

120595119895 119895 isin [1 |B





119895 = 1 so the


119871119896+1(u) and

assume 120588(x y) = 119877 ge 1198711+984858119896




minus ln 10038161003816100381610038161003816120595119895 (x)120595119895 (y)10038161003816100381610038161003816


(119877minus2119862119873119871119896minus2119871119896)(119871119896+1)

]

(60)






120588 (x y) ge 1198711+984858119896minus1

0 lt 120591 lt 120588

1 + 984858 lt 120572

120573 lt 1 minus 120591

(61)


120572 =4

3

120588 =1

6

120591 =1

8

120573 =1

2

(62)




119901 gt 0 there is 1198920= 119892

0(119898 119901 119871


|119892| ge 1198920 any ball B

1198710(119906) and any 119864 isin R

P B1198710(u) 119894119904 (119864119898) -119878 le 119871minus119901

0

P B1198710(u) 119894119904 119898-NL119900119888 le 119871

minus119901

0

(63)




1015840isin (0 1] 1198601015840


119871(u1015840) B(119873)



P dist (Σ119871u1015840 Σ119871u10158401015840) le 119904 le (3119871)

2119873ℎ119871 (2119904) (64)


HB119871(u10158401015840) and

ℎ119871 (119904) = 119862

1015840101584011987111986010158401015840

119904119887and1198871015840

119887 and 1198871015840equiv min [119887 1198871015840] (65)

and 11986010158401015840= 119860

10158401015840(119860



u10158401015840) gt 4119873119871

P exist119864 isin R B119871(u1015840) B

119871(u10158401015840) 119886119903119890 119864-119875119877 le eminus119871

1205731015840

(66)





119873= 4119873 and





H(119873)

B119871(u) = H(1198991015840)

B119871(u1015840)otimes 1(119899

10158401015840)+ 1(119899

1015840)otimesH(119899

10158401015840)

B119871(u10158401015840)(67)

with 1198991015840+11989910158401015840 = 119873 u = (u1015840 u10158401015840) u1015840isinZn1015840

gt u10158401015840isinZn10158401015840

gt while for




(x) ΠJ119888B119871(x)) gt 2119871

(B) If B119871(x) B


then 120588(ΠB119871(x) ΠB

119871(y)) ge 2119871






B(119873)

119871119896(u) = B(119899

1015840)

119871119896(u1015840) times B(119899

10158401015840)

119871119896(u10158401015840) (68)


If the balls B(1198991015840)

119871119896minus1(u1015840) and B(119899

10158401015840)


B(119873)

119871119896(u) is also119898-Loc



HB119871119896 (u)= HB(119899

1015840)119871119896

(u1015840) otimes 1(11989910158401015840)+ 1(119899

1015840)otimesHB(119899

10158401015840)119871119896

(u10158401015840) (69)



119895= 120601

1198951015840 otimes120595

11989510158401015840 where 120601


1015840)119871119896



10158401015840)119871119896

(u10158401015840) Now


119895 (x)Ψ119895 (y)10038161003816100381610038161003816

=100381610038161003816100381610038161206011198951015840 (x1015840)120601

11989510158401015840 (x10158401015840)120595

1198951015840 (y1015840)120595

11989510158401015840 (y10158401015840)10038161003816100381610038161003816

(70)


1015840)

119871119896(u1015840) and B(119899

10158401015840)

119871119896(u10158401015840)


119875 (119899 119896) = 2119873minus119899119901 (1 + 120579)

119896 1 le 119899 le 119873 119896 ge 0 (71)



119871119896(u)

P B(119899)

119871119896(u) 119894119904 119898-NL119900119888 le 119871

minus119875(119899119896)

119896 (72)


119871119896(u) one has

P B(119873)

119871119896(u) 119894119904 119898-NL119900119888 le 2119871

minus2119875(119873119896)

119896le1

4119871minus119875(119873119896)

119896 (73)




1015840)

119871119896(u1015840) B(119899

10158401015840)

119871119896(u10158401015840) is 119898-NLoc


P B(119873)

119871119896(u) is 119898-NL119900119888

le P B(11989910158401015840)

119871119896(u10158401015840) is 119898-NL119900119888

+ P B(11989910158401015840)

119871119896(u10158401015840) is 119898-NL119900119888

le 119871minus119875(1198991015840119896)

119896+ 119871

minus119875(11989910158401015840119896)

119896le 2119871

minus2119875(119873119896)

119896lt1

4119871minus119875(119873119896)

119896

(74)





1198961015840 with 1198961015840 le 119896


119871119896(u)

P B(119873)

119871 119894(u) 119894119904 119898-NL119900119888 le 119871

minus119875(119873119894)

119894 (75)


119871119896+1(x) one has

P B(119873)

119871119896+1(u) 119894119904 119898-NL119900119888 le 119871

minus119875(119873119896+1)

119896+1 (76)

provided that

119901 gt2120572

2

2 minus 1205722119873119889

0 lt 3120579 le min2 minus 1205722

1205722minus2119873119889


(77)


119871119896(u) B(119873)

119871119896(k) one

has for some 119886 119888 gt 0


119871119896(u) B(119873)

119871119896(k) 119886119903119890 (119864119898) -119878

le 119871minus119875(119873119896)

119896equiv 119871

minus119901(1+120599)119896


1+119888119871119896

(78)


N119896+1= B(119873)

119871119896+1(u) is 119898-NLo119888

S(2)

119896


119871119896(u) B(119873)

119871119896(k)

sub B(119873)

119871119896+1(w)

R(2)


119871119896(u) B(119873)

119871119896(k)

sub B(119873)

119871119896+1(w)

(79)


N119896+1sub S

(2)

119896subR

(2)

119896cup (S

(2)

119896R

(2)

119896) (80)


119896 le eminus119871

1205731015840


119896R

(2)


119871119896+1(w) and


S(2)

119896(u k) = exist119864 B(119873)

119871119896(u) B(119873)

119871119896(k) are (119864119898) -S (81)


119896(u k) R(2)

119896 either B(119873)

119871119896(u) or B(119873)

119871119896(k)


119871119896(u) is 119864-CNR



pair of 119898-NL119900119888 balls B(119873)

119871119896minus1(y1015840) B(119873)


(12)1198711+120575


(1) Both B(119873)

119871119896minus1(y1015840) and B(119873)

119871119896minus1(y10158401015840) are FI


120588 (ΠB119871119896minus1(u) ΠB

119871119896minus1(k)) ge 2119871

119896 (82)


119896minus1= 119871

11205722

119896+1and 119871

0large enough

P B(119873)

119871119896minus1(y1015840) B(119873)

119871119896minus1(y10158401015840) are 119898-NL119900119888

le 2119871minus(2119901120572

2)(1+120579)

119896minus1

119896+1

(83)


119871119896(w)|2 le (321198732)1198712119873


P S(2)

119896R

(2)

119896 le32119873

2119871minus(2119901120572

2)(1+120579)

119896minus1+2119873

119896+1 (84)



119896+1 under con-


2119901

12057222119873minus119873119901 (1 + 120579)

119896minus1minus 2119873119889 gt 2

119873minus119873119901 (1 + 120579)

119896+1 (85)

or equivalently

(1 + 120579)2lt2119901

1205722minus

2119873119889

119901 (1 + 120579)119896minus1 (86)


3120599 le2 minus 120572

2

1205722minus2119873

119901 (87)


P S(2)

119896 le P R

(2)

119896 + P S

(2)

119896R

(2)

119896

le eminus1198711205731015840

119896 +1

4119871minus119875(119873119896+1)

119896+1lt1

2119871minus119875(119873119896+1)

119896+1

(88)


119871119896minus1(y1015840) orB(119873)


P B(119873)

119871119896minus1(y1015840) B(119873)

119871119896minus1(y10158401015840) is 119898-NL119900119888

le min (P B(119873)

119871119896minus1(y1015840) is 119898-NL119900119888

P B(119873)

119871119896minus1(y10158401015840) is 119898-NL119900119888) le 119871

minus2119901(1+120579)119896minus1

119896minus1

lt 119871minus119901(1+120579)

119896+1

119896+1

(89)



119871minus119875(119873119896)

119896equiv 119871

minus119901(1+120599)119896


1+119888119871119896 (90)




P B(119873)

119871119896(u) 119894119904 119898-NL119900119888 le

1

4119871minus119901(1+120579)

119896

119896(91)


minus119901(1+120579)119896


119895of operator HB(119873)119871119896 (u)



119896one

has10038161003816100381610038161003816Ψ119895(x) Ψ






1003816100381610038161003816Ψ (y)1003816100381610038161003816 le eminus119898y

(93)








119871119896(y) sub

B119871119896+2(z) Fix 120596 isin Ω1015840


2le 1 implies Ψ

infinle 1 there is


119899isin B

119871119896minus1(0) then






119871119896+2(z) with 120588(y z) isin [119871

119896+1 119871

119896+2)


1+120575

119896) is (120582119898)-NS Note that



119877(y) with

119877 = 120588(y z)minus21198711+120575119896minus1 gt (12)119871

1+2120575



minusln 1003816100381610038161003816Ψ (y)

1003816100381610038161003816

120588 (y z)ge 119898 (1 + 119871

minus120591

119896) (1 minus

2119871119896+ 1

120588 (y z))

ge 119898(1 +1

2119871minus120591

119896) gt 119898

(95)









119871(119909) B


P exist 119864 isin 119868 B119871(x) B

119871(y) 119886119903119890 (119864119898) -S le 119891 (119871) (96)


119871(119909) cup B

119871(119910) one has

E[ sup120601isinB1(119868)

10038161003816100381610038161003816⟨1x1003816100381610038161003816120601 (HΛ (120596))

1003816100381610038161003816 1y⟩10038161003816100381610038161003816]

le 119862119871119889119890minus119898119871+ 119891 (119871)

(97)




Λ sup B119871(119909) cup B

119871(119910)


10038161003816100381610038161003816⟨1x1003816100381610038161003816120601 (HΛ (120596))

1003816100381610038161003816 1y⟩10038161003816100381610038161003816] le 119862119890

minus119886 ln1+119888120588(xy) (98)










same 119888 119886 gt 0 as in (98)

E[ sup120601isinC1(119868)

10038161003816100381610038161003816⟨1x1003816100381610038161003816120601 (H (120596))

1003816100381610038161003816 1y⟩10038161003816100381610038161003816] le 119862eminus119886 ln

1+119888120588(xy) (99)




int120601 (120582) 119889120583xyB120596 (120582)

= ⟨1x1003816100381610038161003816 120601 (HB (120596))Π119868

(HB (120596))100381610038161003816100381610038161y⟩

(100)








119899rarr H with







119871119896

E [1003816100381610038161003816120583

xy120596


E [100381610038161003816100381610038161003816120583xyB119871119896

100381610038161003816100381610038161003816(E)] (101)




120601119905 120582 997891997888rarr 119891

119868(120582) ei119905120582 119905 isin R (102)








984858 = 2120575

120572 = 1 + 4120575

120591 =120575

2

(103)



120574 (119898 119871 119899) = 119898 (1 + 119871minus120591)119873minus119899+1

(104)



120588(x y) ge 1198621198731198711+120575 with 119862

119873= 4119873






119871(x) admits

a factorization

B(119873)

119871(x) = B(119899

1015840)

119871(x1015840) times B(119899

10158401015840)

119871(x10158401015840) (105)


119871(x) B


then 120588(ΠB119871(x) ΠB

119871(y)) ge 1198711+120575




119877119877 ge 119903



119880(2)

119877(119903) = 1

119903le119877119880

(2)(119903) (106)


gt



119877(x1015840) + U



11987110158401015840(x10158401015840) with

120588(B1198711015840(x1015840)B



H(119877)

B = H(119877)

B1198711015840(x1015840) otimes 1

(11989910158401015840)+ 1(119899

1015840)otimesH(119877)

B11987110158401015840

(x10158401015840) (107)


B(119873)


119873= 4119898119873


119871(x) =

B1198711015840(x1015840) times B

11987110158401015840(x10158401015840) with dist(x1015840 x10158401015840) gt (119862

1198732119873)119871 = 2119898119871

so that with 119877119896= (119862

1198732119873)119871

119896 one has

120598 (119877119896) lt eminus2119898119871119896 lt

1

2eminus119871120573

119896 (108)


119871119896(u) =

B(1198991015840)

119871119896(uJ)timesB

(11989910158401015840)



(a) dist(x1015840 x10158401015840) gt 119877119896= (119862

1198732119873)119871

119896

(b) B(1198991015840)

119871119896(u1015840) and B(119899

10158401015840)

119871119896(u10158401015840) are119898-Loc

(c) B(119873)

119871119896(u) is 119864-NR


119871119896(u) is (119864119898)-NS

Proof Set 119877119896= (119862

1198732119873)119871


above Denote

H(119877119896119873)

u119896 = H(119877119896)

B(119873)119871119896 (u)

G (119864) = (HB(119873)119871119896 (u)minus 119864)

minus1

(109)

and G(119877119896)(119864) = (H(119877119896119873)



H(119877119896 119873)

u119896 = HB(1198991015840)119871119896

(u1015840) otimes 1(11989910158401015840)+ 1(119899

1015840)otimesHB(119899

1015840)119871119896

(x10158401015840) (110)

with u = (u1015840 u10158401015840) isin Z119889119899

1015840

times Z11988911989910158401015840


119886119887= 120582

119886+120583

119887 where 120582

119886 = Σ(HB119871119896 (u



119887 = Σ(HB119871119896 (u


of H(119877119896 119873)


119886otimes 120595

119887 where


1015840)119871119896


of HB(11989910158401015840)119871119896


119886+ 120583

119887




|119864 minus (120582119886+ 120583

119887)| ge e119871

120573


119887)| ge e119871

120573

119896 and alsoas |(119864 minus 120583

119887) minus 120582

119886)| ge e119871

120573


G(119877119896) (u y 119864)

= sum

120582119886

sum

120583119887

120601119886(u1015840)120601

119886(y1015840)120595

119887(u10158401015840)120595

119887(y10158401015840)

(120582119886+ 120583

119887) minus 119864

= sum

120582119886

P1015840

119886(u1015840 y1015840)G(119877119896)

B119871119896 (u10158401015840)(u10158401015840 y10158401015840 119864 minus 120582

119886)

= sum

120583119887

P10158401015840

119887(u10158401015840 y10158401015840)G(119877119896)

B119871119896 (u1015840)(u1015840 y1015840 119864 minus 120583

119887)

(111)


B119871119896 (u1015840)(119864minus120583

119887) andG(119877119896)

B119871119896 (u10158401015840)(119864minus120582

119886) are

nonresonant1003817100381710038171003817100381710038171003817G(119877119896)

B119871119896 (u1015840)(119864 minus 120583

119887)

1003817100381710038171003817100381710038171003817le e119871

120573

119896

1003817100381710038171003817100381710038171003817G(119877119896)

B119871119896 (u10158401015840)(119864 minus 120582

119886)

1003817100381710038171003817100381710038171003817le e119871

120573

119896

(112)


y1015840) = 119871119896 in which case we

infer from (b)

10038161003816100381610038161003816G(119877119896) (u y 119864)10038161003816100381610038161003816 le

10038161003816100381610038161003816B119871119896(u10158401015840)10038161003816100381610038161003816eminus120574(119898119871119896119873minus1)119871119896+119871

120573

119896 (113)

or 120588(u10158401015840 y10158401015840) = 119871


10038161003816100381610038161003816G(119877119896) (u y 119864)10038161003816100381610038161003816 le

10038161003816100381610038161003816B119871119896(u1015840)10038161003816100381610038161003816eminus120574(119898119871119896119873minus1)119871119896+119871

120573

119896 (114)



119871119896+ 119871

120573

119896+ 119862 ln 119871

119896)

lt1

2eminus120574(119898119871119896119873)

(115)




10038171003817100381710038171003817G minus G(119877119896)

10038171003817100381710038171003817le10038171003817100381710038171003817U minus U(119877119896)

10038171003817100381710038171003817

10038171003817100381710038171003817G(119877119896)

10038171003817100381710038171003817G le eminus2119898119871119896e2119871

120573

119896

le1

2eminus120574(119898119871119896119873)

(117)


10038161003816100381610038161003816G (x y 119864) minus G(119877119896) (x y 119864)10038161003816100381610038161003816 le

1

2eminus120574(119898119871119896 119873)


119871119896+1(u)

(118)

Lemma 33 Let B119871119896+1(x)B



P B119871119896+1(x) B

119871119896+1(y) 119886119903119890 (119864119898) -119878

lt1

4119871minus119875(119873119896+1)

119896+1

(119)

Proof Let

R(2)= exist119864 B

119871119896+1(x) B

119871119896+1(y) 119886119903119890 119864-R

S(2)= B

119871119896+1(x) B119871119896+1

(y) 119886119903119890 (119864119898) -S (120)

ThenPS(2) le PR(2)

+PS(2)R(2)


119871119896+1(x) B

119871119896+1(y) must be 119864-NR


B(119873)

119871119896+1(x) = B(119899

1015840)

119871119896+1(x1015840) times B(119899

1015840)

119871119896+1(x10158401015840) (121)


119871119896+1(x10158401015840) are119898-Loc and

B119871119896+1(x) is (119864119898)-S then B




B(119873)

119871119896+1(y) = B(119899

1015840)

119871119896+1(y1015840) times B(119899

10158401015840)

119871119896+1(y10158401015840) (122)

If both B(1198991015840)119871119896+1(y1015840) and B(11989910158401015840)

119871119896+1(y10158401015840) are 119898-Loc and

B(119873)

119871119896+1(y) is (119864119898)-S then B(119873)

119871119896+1(y)must be 119864-R Let


1015840)

119871119896+1(x1015840) B(119899

10158401015840)

119871119896+1(x10158401015840)

B(1198991015840)

119871119896+1(y1015840) B(119899

1015840)

119871119896+1(y10158401015840) is 119898-NL119900119888

(123)


P L(4) le 4119871

minus119875(119873minus1119896+1)

119896+1= 4119871

minus2119875(119873119896+1)

119896+1 (124)



large enough

P S(2) le P R

(2) + P L

(4)

le eminus119871120573

119896 + 4119871minus2119875(119873119896+1)

119896+1lt1

4119871minus119875(119873119896+1)

119896+1

(125)






B119871119896(k) sub B

119871119896+1(u)with 120588(w k) ge 119862

1198731198711+120575

119896is (119864119898)-NS In the







119896also large)

1198711+984858

119896≫ 119871

1+120575


120588 (xB(119862119873minus1)119871119896

(w)) + 120588 (yB(119862119873minus1)119871119896(w))

ge 119877 minus (2119862119873minus 2) 119871

1+120575

119896

(126)

Assume first that

1199031015840= 120588 (xB

(119862119873minus1)1198711+120575119896(w)) ge 119871

119896+ 1

11990310158401015840= 120588 (yB

(119862119873minus1)1198711+120575119896(w)) ge 119871119896 + 1

(127)


1199031015840(x) and in B

11990310158401015840(y) are


1)1198711+120575

119896minus 2 ge 119877 minus 2119862

1198731198711+120575

119896










(119877minus21198621198731198711+120575119896 minus2119871119896)(119871119896+1)

sdot e119871120573

119896+1] = 119898(1 +1

2119871minus120591

119896)

119869119871119896

119871119896+ 1119877(1 minus

3119862119873

119877

sdot 119871119896) minus 119871

120573

119896+1= 119898119877[(1 +

1

2119871minus120591

119896)

119869


119896)


119873119871119896

1198711+984858

119896

) minus119871120573

119896+1

119898119877] = 119898119877[(1 +

1

2119871minus120591

119896)

119869


119896) (1 minus 3119862

119873119871minus984858

119896) minus

119871120573

119896+1

1198981198711+984858

119896

] ge 119898(1 +1

4


119896)

119869

119877 ge 120574 (119898 119871119896+1) 119877

(128)






1 119909

119873) 119909





1003816100381610038161003816 le 119862119889119871119889 (129)





by the functions

Φa =1

radic119873

sum

120587isinS119873

119873

⨂

119895=1

1119886120587minus1(119895) (130)

where a = 1198861 119886

119873 with 119886

1 119886

119873 = 119873



set is

ZNgt

= a = 1198861 119886

119873 119886

119895isinZ 119886

1 119886

119873 = 119873

(131)



1 119888

2 119888

119873 b = 119887

1 119888

2 119888

119873 with

1198861 119887

1 119888

2 119888

119873 = 119873 + 1

(ii) dZ(1198861 1198871) = 1


120588 (x y) = min120587isinS119873

max1le119895le119873

dZ (119909120587minus1(119895) 119910119895) (132)




119892V(x 120596) + U(x) where

U (x) = sum119894 =119895

119880(d (119909119894 119909

119895)) 119880 N 997888rarr R (133)


V (x 120596) = 119881 (1199091 120596) + sdot sdot sdot + 119881 (119909

119873 120596) (134)






Appendix



V (1199091 119909

119873 120596) = sum

119910isinxn119910119881 (119910 120596) (A1)





119908(y)) 119881 (119908 120596)

+ sum

V =119908

119888V119881 (V 120596) (A2)


119888V = nV (x) minus nV (y) (A3)



119908 while

n119908(x) minus n

119908(y) = 119888

119908isin Z 0 hence 1003816100381610038161003816119888119908

1003816100381610038161003816 ge 1 (A4)

Therefore

P 1003816100381610038161003816119892V (x 120596) minus 119892V (y 120596)

1003816100381610038161003816 le 119904

= E [P 1003816100381610038161003816V (x 120596) minus V (y 120596)

1003816100381610038161003816 le10038161003816100381610038161198921003816100381610038161003816

minus1119904 | F

=119908]

= E [P 1003816100381610038161003816119888119908119881 (119908 120596) | z (120596)

1003816100381610038161003816 le10038161003816100381610038161198921003816100381610038161003816

minus1119904 | F

=119908]

(A5)




120596 119888119908119881 (119908 120596) isin [z (120596) minus

10038161003816100381610038161198921003816100381610038161003816

minus1119904 z (120596) +

10038161003816100381610038161198921003816100381610038161003816

minus1119904]

= 120596 119881 (119908 120596)

isin [z (120596) minus11990410038161003816100381610038161198921003816100381610038161003816

minus1

119888119908

z (120596) +11990410038161003816100381610038161198921003816100381610038161003816

minus1

119888119908

]

(A6)


P 1003816100381610038161003816V (x 120596) minus V (y 120596)

1003816100381610038161003816 le10038161003816100381610038161198921003816100381610038161003816

minus1119904

le sup119886isinR

(119865119881119908(119886 + 2119904) minus 119865

119881119908(119886))

(A7)


at 119908 given F=119908 Since 119865




1003816100381610038161003816 le 119904 (A8)


1198710(u) admits



0 To prove the




100381610038161003816100381610038161003816GB1198710 (u)

(x y 119864)100381610038161003816100381610038161003816le eminus119898(119904)d(xy)

(A9)





119899

119895=11198612119871(119909

119895) If




1198711+120575(x) can be


(⋃

119895isinJ

1198612119871(119909

119895))∐(⋃

119894isinJ119888

1198612119871(119909

119894)) (A10)


120588 (119861119871(119909

119894) 119861

119871(119909

119895)) gt 2119871 (A11)


119873119871 then for some 119895

∘ 120588(119909

119895∘ 119910

119895∘) gt 119862

119873119871


120588 (119909119895 119909

119895∘) le diamΠx le 119860

119873119871

120588 (119910119895 119910

119895∘) le diamΠy lt 119860

119873119871

(A12)


120588 (119909119894 119910

119895) ge 120588 (119909

119895∘ 119910

119895∘) minus 120588 (119909

119894 119909

119895∘) minus 120588 (119910

119895 119910

119895∘)

ge 119862119873119871 minus 2119860

119873119871 ge 3119871

(A13)


yielding

120588 (Π119861119871(x) Π119861

119871(y)) gt 3119871 minus 2119871 = 119871 (A14)



Acknowledgments


References

































Volume 2014




Journal of











Journal of


Function Spaces






Algebra














120585119876(120596) = |119876|

minus1sum

119909isin119876

119881 (119909 120596) (23)


119876(120596)119909 isin 119876 Denote byF


generated by 120578119909 119881

119910 119909 isin 119876 119910 notin 119876


(W3) There exist1198621015840 119860

1015840 119887




P 120585119876(120596) isin [120582 (120596) 120582 (120596) + 119904] le 119862

1015840|119876|

1198601015840

1199041198871015840

(24)









1lowast 120583

2 where 120583

2is



0 lt 120588lowastle 120588

1 (119905) le 120588 lt infin

1205881015840

1(119905)

1205881(119905)le 119862

1lt infin

forall119905 isin (119886 119887)

(25)




119880 (119903) = 0 forall119903 ge 1199030 (26)


|119880 (119903)| le 119862eminus1198881199031minus120577

forall119903 ge 0 (27)


26 Main Results








119895 x)

large enough

10038161003816100381610038161003816Ψ

119895(x 120596)10038161003816100381610038161003816 le eminus119898120588(x119895x) 119898 gt 0 (28)


E[sup119905isinR

10038161003816100381610038161003816⟨1y10038161003816100381610038161003816eminusi119905H(120596)10038161003816100381610038161003816

1x⟩10038161003816100381610038161003816] le 119862eminus119886 ln

1+119888120588(xy) (29)


E[sup119905isinR

100381710038171003817100381710038171003817e(1198862) ln








H(120596) on 119868




119873

y = 1199101 119910

119873


dZ (119909119894 y) max119894

dZ (119910119894 x)] (31)

where

dZ (119909 y) = min119895

dZ (119909 119910119895) (32)


E[sup119905isinR

10038161003816100381610038161003816⟨1y10038161003816100381610038161003816eminusi119905H(120596)10038161003816100381610038161003816

1x⟩10038161003816100381610038161003816] le 119862 (x) eminus119886 ln

1+119888dH(xy) (33)



119871(119904) le 119862119871

119860119904119899 119862119860 isin

(0 +infin) and 119887 isin (0 1]





gt |Λ| equiv


HΛ (120596) = 1

ΛH (120596) 1Λ HΛ (34)







(Γxy119891) (x) = 120575xy119891 (y) (35)


Λ(119864) = (H

Λminus119864)




10038161003816100381610038161003816

le 119862ℓ


10038161003816100381610038161003816GBℓ(x) (x k)

10038161003816100381610038161003816max

k1015840isin120597+Bℓ(x)

10038161003816100381610038161003816GB119871 (k

1015840 y)10038161003816100381610038161003816

(36)

where

119862ℓ=1003816100381610038161003816120597Bℓ

(k)1003816100381610038161003816 le 119862 (119873) ℓ119873 (37)


10038161003816100381610038161003816G

119861119871(x y)10038161003816100381610038161003816

le (119862ℓmax

120588(xk)=ℓ

10038161003816100381610038161003816GBℓ(x) (x k)

10038161003816100381610038161003816) max

120588(uk1015840)leℓ+1

10038161003816100381610038161003816GB119871 (k

1015840 y)10038161003816100381610038161003816

(38)


10038161003816100381610038161003816GB119871 (x y 119864)

10038161003816100381610038161003816

le 119862ℓ

10038171003817100381710038171003817GBℓ(u) (119864)

10038171003817100381710038171003817max

k120588(uk)leℓ+1

10038161003816100381610038161003816GB119871 (k y 119864)

10038161003816100381610038161003816

(39)


ℓ(u)

1003816100381610038161003816120595 (x)1003816100381610038161003816 le 119862ℓ

10038171003817100381710038171003817GBℓ(x) (119864)

10038171003817100381710038171003817max

y120588(xy)leℓ+11003816100381610038161003816120595 (y)

1003816100381610038161003816 (40)



gt


119877(119906) rarr R



ℓ(119909) sub Λ

119877(119906) one has

119891 (119909) le 119902 max119910d(119909119910)leℓ

119891 (119910) (41)





M (119891 119861)

le 119902(119871minusℓ)(ℓ+1)

M (119891 119861)

M (119891 Λ) = max119909isinΛ

1003816100381610038161003816119891 (119909)1003816100381610038161003816

(42)





M (119891 119861)

(43)





1015840) ⊊ Λ and in 11990910158401015840 isin 119861

11990310158401015840(119906

10158401015840) ⊊ Λ with

d(1199061015840 11990610158401015840) ge 1199031015840 + 11990310158401015840 + 2 Then

119891 (1199061015840 119906

10158401015840) le 119902


10158401015840+1)(ℓ+1)rfloor

M (119891 119861)

le 119902(1199031015840+11990310158401015840minus2ℓ)(ℓ+1)

M (119891 119861)

(44)



tion

11989111991010158401015840 119910

1015840997891997888rarr 119891 (119910

1015840 119910

10158401015840) (45)


1015840) ⊊ Λ so by Lemma 6

11989111991010158401015840 (119906

1015840) = 119891 (119906

1015840 119910

10158401015840) le 119902

lfloor(1199031015840+1)(ℓ+1)rfloor

M (119891 Λ times Λ) (46)


10158401015840997891rarr 119891(119906

1015840 119910

10158401015840) which is

(ℓ 119902)-dominated in 11986111990310158401015840(119906






119896+1=

lceil119871120572

119896rceil 119871



0to be large enough


119871(u) is called




ℓ(u) sube B

119871(u) with ℓ ge 1198711120572





10038161003816100381610038161003816GB119871(u) (u y 119864 120596)

10038161003816100381610038161003816le eminus120574(119898119871)119871+2119871

120573

(47)

where

120574 (119898 119871) = 119898 (1 + 119871minus120591) 120591 =

1

8 (48)




120595119895 119895 = 1 |B



0isin B

119871119896(u)withd(x

0 y

0) gt 119871

119896minus1+1

and an integer

119877 isin [119871119896minus1 d (x

0 y

0) minus 119871

119896minus1+ 1] (49)


119877(x

0) is (119864119898)-NS and set

119902 = eminus120574(119898ℓ)ℓ (50)

Then



120595119895=

|120595119895⟩⟨120595





B119877(x

0) ni x 997891997888rarr

100381610038161003816100381610038161003816GB119871119896 (u)

(x y 119864)100381610038161003816100381610038161003816

(51)


0) and globally



119877(y


0fixed




satisfies10038161003816100381610038161003816120595

119895(x)120595

119895(y)10038161003816100381610038161003816 le eminus120574(119898119871)120588(xy) (52)




120588(x y) ge 119862119873119871 with 119862

119873= 2119860

119873+ 3 119860

119873= 4119873



in Section 43)



ℓ(y) and119898-







1003816100381610038161003816GB (x y 119864)1003816100381610038161003816 le sum

119864119895isinΣ(HB)

10038161003816100381610038161003816120595

119895(x)1003816100381610038161003816100381610038161003816100381610038161003816120595

119895(y)10038161003816100381610038161003816

10038161003816100381610038161003816119864 minus 119864

119895

10038161003816100381610038161003816

(53)


120573


1003816100381610038161003816GB (x y 119864)1003816100381610038161003816 le eminus120574(119898119871)119871+119871

120573+ln|B|

le eminus120574(119898119871)119871+2119871120573

(54)



119898(1 + 119871minus120591

119896) 119871

119896minus 2119871

120573

119896= 119898119871

119896+ 119898119871

1minus120591

119896minus 2119871

120573

119896

ge 119898(1 +1

2119871minus120591

119896) 119871

119896

(55)




(1)




(119864119898)-NS(B) if a ball B


(119864119898)-NS




119871119896(k) sub B

119871119896(u) with






120588 (xB(119862119873minus1)119871119896

(w)) + 120588 (yB(119862119873minus1)119871119896(w))

ge 119877 minus (2119862119873minus 2) 119871

119896

(56)

Assume first that

1199031015840= 120588 (xB

(119862119873minus1)119871119896(w)) ge 119871

119896+ 1

11990310158401015840= 120588 (yB

(119862119873minus1)119871119896(w)) ge 119871

119896+ 1

(57)



119896are auto-


Furthermore

1199031015840+ 119903

10158401015840ge 119877 minus 2 (119862

119873minus 1) 119871

119896minus 2 ge 119877 minus 2119862

119873119871119896 (58)










(119877minus2119862119873119871119896minus2119871119896)(119871119896+1)

e119871120573

119896+1]

= 119898(1 +1

2119871minus120591

119896)119871119896

119871119896+ 1119877 (1 minus 3C

119873119877minus1119871119896) minus 119871

120573

119896+1

= 119898119877[(1 +1

2119871minus120591

119896) (1 minus 119871

minus1

119896) (1 minus 3119862

119873119871minus984858

119896)

minus119871120573

119896+1

119898119877] ge 119898(1 +

1

4119871minus120591

119896)119877 ge 120574 (119898 119871

119896+1) 119877

(59)




B119871119896(x) B


119871119896+1(u) is 119864-


119871119896(y) must


the hypothesis






119895 (k1015840 k10158401015840) 997891rarr |120595

119895(k1015840)120595

119895(k10158401015840)| where

120595119895 119895 isin [1 |B





119895 = 1 so the


119871119896+1(u) and

assume 120588(x y) = 119877 ge 1198711+984858119896




minus ln 10038161003816100381610038161003816120595119895 (x)120595119895 (y)10038161003816100381610038161003816


(119877minus2119862119873119871119896minus2119871119896)(119871119896+1)

]

(60)






120588 (x y) ge 1198711+984858119896minus1

0 lt 120591 lt 120588

1 + 984858 lt 120572

120573 lt 1 minus 120591

(61)


120572 =4

3

120588 =1

6

120591 =1

8

120573 =1

2

(62)




119901 gt 0 there is 1198920= 119892

0(119898 119901 119871


|119892| ge 1198920 any ball B

1198710(119906) and any 119864 isin R

P B1198710(u) 119894119904 (119864119898) -119878 le 119871minus119901

0

P B1198710(u) 119894119904 119898-NL119900119888 le 119871

minus119901

0

(63)




1015840isin (0 1] 1198601015840


119871(u1015840) B(119873)



P dist (Σ119871u1015840 Σ119871u10158401015840) le 119904 le (3119871)

2119873ℎ119871 (2119904) (64)


HB119871(u10158401015840) and

ℎ119871 (119904) = 119862

1015840101584011987111986010158401015840

119904119887and1198871015840

119887 and 1198871015840equiv min [119887 1198871015840] (65)

and 11986010158401015840= 119860

10158401015840(119860



u10158401015840) gt 4119873119871

P exist119864 isin R B119871(u1015840) B

119871(u10158401015840) 119886119903119890 119864-119875119877 le eminus119871

1205731015840

(66)





119873= 4119873 and





H(119873)

B119871(u) = H(1198991015840)

B119871(u1015840)otimes 1(119899

10158401015840)+ 1(119899

1015840)otimesH(119899

10158401015840)

B119871(u10158401015840)(67)

with 1198991015840+11989910158401015840 = 119873 u = (u1015840 u10158401015840) u1015840isinZn1015840

gt u10158401015840isinZn10158401015840

gt while for




(x) ΠJ119888B119871(x)) gt 2119871

(B) If B119871(x) B


then 120588(ΠB119871(x) ΠB

119871(y)) ge 2119871






B(119873)

119871119896(u) = B(119899

1015840)

119871119896(u1015840) times B(119899

10158401015840)

119871119896(u10158401015840) (68)


If the balls B(1198991015840)

119871119896minus1(u1015840) and B(119899

10158401015840)


B(119873)

119871119896(u) is also119898-Loc



HB119871119896 (u)= HB(119899

1015840)119871119896

(u1015840) otimes 1(11989910158401015840)+ 1(119899

1015840)otimesHB(119899

10158401015840)119871119896

(u10158401015840) (69)



119895= 120601

1198951015840 otimes120595

11989510158401015840 where 120601


1015840)119871119896



10158401015840)119871119896

(u10158401015840) Now


119895 (x)Ψ119895 (y)10038161003816100381610038161003816

=100381610038161003816100381610038161206011198951015840 (x1015840)120601

11989510158401015840 (x10158401015840)120595

1198951015840 (y1015840)120595

11989510158401015840 (y10158401015840)10038161003816100381610038161003816

(70)


1015840)

119871119896(u1015840) and B(119899

10158401015840)

119871119896(u10158401015840)


119875 (119899 119896) = 2119873minus119899119901 (1 + 120579)

119896 1 le 119899 le 119873 119896 ge 0 (71)



119871119896(u)

P B(119899)

119871119896(u) 119894119904 119898-NL119900119888 le 119871

minus119875(119899119896)

119896 (72)


119871119896(u) one has

P B(119873)

119871119896(u) 119894119904 119898-NL119900119888 le 2119871

minus2119875(119873119896)

119896le1

4119871minus119875(119873119896)

119896 (73)




1015840)

119871119896(u1015840) B(119899

10158401015840)

119871119896(u10158401015840) is 119898-NLoc


P B(119873)

119871119896(u) is 119898-NL119900119888

le P B(11989910158401015840)

119871119896(u10158401015840) is 119898-NL119900119888

+ P B(11989910158401015840)

119871119896(u10158401015840) is 119898-NL119900119888

le 119871minus119875(1198991015840119896)

119896+ 119871

minus119875(11989910158401015840119896)

119896le 2119871

minus2119875(119873119896)

119896lt1

4119871minus119875(119873119896)

119896

(74)





1198961015840 with 1198961015840 le 119896


119871119896(u)

P B(119873)

119871 119894(u) 119894119904 119898-NL119900119888 le 119871

minus119875(119873119894)

119894 (75)


119871119896+1(x) one has

P B(119873)

119871119896+1(u) 119894119904 119898-NL119900119888 le 119871

minus119875(119873119896+1)

119896+1 (76)

provided that

119901 gt2120572

2

2 minus 1205722119873119889

0 lt 3120579 le min2 minus 1205722

1205722minus2119873119889


(77)


119871119896(u) B(119873)

119871119896(k) one

has for some 119886 119888 gt 0


119871119896(u) B(119873)

119871119896(k) 119886119903119890 (119864119898) -119878

le 119871minus119875(119873119896)

119896equiv 119871

minus119901(1+120599)119896


1+119888119871119896

(78)


N119896+1= B(119873)

119871119896+1(u) is 119898-NLo119888

S(2)

119896


119871119896(u) B(119873)

119871119896(k)

sub B(119873)

119871119896+1(w)

R(2)


119871119896(u) B(119873)

119871119896(k)

sub B(119873)

119871119896+1(w)

(79)


N119896+1sub S

(2)

119896subR

(2)

119896cup (S

(2)

119896R

(2)

119896) (80)


119896 le eminus119871

1205731015840


119896R

(2)


119871119896+1(w) and


S(2)

119896(u k) = exist119864 B(119873)

119871119896(u) B(119873)

119871119896(k) are (119864119898) -S (81)


119896(u k) R(2)

119896 either B(119873)

119871119896(u) or B(119873)

119871119896(k)


119871119896(u) is 119864-CNR



pair of 119898-NL119900119888 balls B(119873)

119871119896minus1(y1015840) B(119873)


(12)1198711+120575


(1) Both B(119873)

119871119896minus1(y1015840) and B(119873)

119871119896minus1(y10158401015840) are FI


120588 (ΠB119871119896minus1(u) ΠB

119871119896minus1(k)) ge 2119871

119896 (82)


119896minus1= 119871

11205722

119896+1and 119871

0large enough

P B(119873)

119871119896minus1(y1015840) B(119873)

119871119896minus1(y10158401015840) are 119898-NL119900119888

le 2119871minus(2119901120572

2)(1+120579)

119896minus1

119896+1

(83)


119871119896(w)|2 le (321198732)1198712119873


P S(2)

119896R

(2)

119896 le32119873

2119871minus(2119901120572

2)(1+120579)

119896minus1+2119873

119896+1 (84)



119896+1 under con-


2119901

12057222119873minus119873119901 (1 + 120579)

119896minus1minus 2119873119889 gt 2

119873minus119873119901 (1 + 120579)

119896+1 (85)

or equivalently

(1 + 120579)2lt2119901

1205722minus

2119873119889

119901 (1 + 120579)119896minus1 (86)


3120599 le2 minus 120572

2

1205722minus2119873

119901 (87)


P S(2)

119896 le P R

(2)

119896 + P S

(2)

119896R

(2)

119896

le eminus1198711205731015840

119896 +1

4119871minus119875(119873119896+1)

119896+1lt1

2119871minus119875(119873119896+1)

119896+1

(88)


119871119896minus1(y1015840) orB(119873)


P B(119873)

119871119896minus1(y1015840) B(119873)

119871119896minus1(y10158401015840) is 119898-NL119900119888

le min (P B(119873)

119871119896minus1(y1015840) is 119898-NL119900119888

P B(119873)

119871119896minus1(y10158401015840) is 119898-NL119900119888) le 119871

minus2119901(1+120579)119896minus1

119896minus1

lt 119871minus119901(1+120579)

119896+1

119896+1

(89)



119871minus119875(119873119896)

119896equiv 119871

minus119901(1+120599)119896


1+119888119871119896 (90)




P B(119873)

119871119896(u) 119894119904 119898-NL119900119888 le

1

4119871minus119901(1+120579)

119896

119896(91)


minus119901(1+120579)119896


119895of operator HB(119873)119871119896 (u)



119896one

has10038161003816100381610038161003816Ψ119895(x) Ψ






1003816100381610038161003816Ψ (y)1003816100381610038161003816 le eminus119898y

(93)








119871119896(y) sub

B119871119896+2(z) Fix 120596 isin Ω1015840


2le 1 implies Ψ

infinle 1 there is


119899isin B

119871119896minus1(0) then






119871119896+2(z) with 120588(y z) isin [119871

119896+1 119871

119896+2)


1+120575

119896) is (120582119898)-NS Note that



119877(y) with

119877 = 120588(y z)minus21198711+120575119896minus1 gt (12)119871

1+2120575



minusln 1003816100381610038161003816Ψ (y)

1003816100381610038161003816

120588 (y z)ge 119898 (1 + 119871

minus120591

119896) (1 minus

2119871119896+ 1

120588 (y z))

ge 119898(1 +1

2119871minus120591

119896) gt 119898

(95)









119871(119909) B


P exist 119864 isin 119868 B119871(x) B

119871(y) 119886119903119890 (119864119898) -S le 119891 (119871) (96)


119871(119909) cup B

119871(119910) one has

E[ sup120601isinB1(119868)

10038161003816100381610038161003816⟨1x1003816100381610038161003816120601 (HΛ (120596))

1003816100381610038161003816 1y⟩10038161003816100381610038161003816]

le 119862119871119889119890minus119898119871+ 119891 (119871)

(97)




Λ sup B119871(119909) cup B

119871(119910)


10038161003816100381610038161003816⟨1x1003816100381610038161003816120601 (HΛ (120596))

1003816100381610038161003816 1y⟩10038161003816100381610038161003816] le 119862119890

minus119886 ln1+119888120588(xy) (98)










same 119888 119886 gt 0 as in (98)

E[ sup120601isinC1(119868)

10038161003816100381610038161003816⟨1x1003816100381610038161003816120601 (H (120596))

1003816100381610038161003816 1y⟩10038161003816100381610038161003816] le 119862eminus119886 ln

1+119888120588(xy) (99)




int120601 (120582) 119889120583xyB120596 (120582)

= ⟨1x1003816100381610038161003816 120601 (HB (120596))Π119868

(HB (120596))100381610038161003816100381610038161y⟩

(100)








119899rarr H with







119871119896

E [1003816100381610038161003816120583

xy120596


E [100381610038161003816100381610038161003816120583xyB119871119896

100381610038161003816100381610038161003816(E)] (101)




120601119905 120582 997891997888rarr 119891

119868(120582) ei119905120582 119905 isin R (102)








984858 = 2120575

120572 = 1 + 4120575

120591 =120575

2

(103)



120574 (119898 119871 119899) = 119898 (1 + 119871minus120591)119873minus119899+1

(104)



120588(x y) ge 1198621198731198711+120575 with 119862

119873= 4119873






119871(x) admits

a factorization

B(119873)

119871(x) = B(119899

1015840)

119871(x1015840) times B(119899

10158401015840)

119871(x10158401015840) (105)


119871(x) B


then 120588(ΠB119871(x) ΠB

119871(y)) ge 1198711+120575




119877119877 ge 119903



119880(2)

119877(119903) = 1

119903le119877119880

(2)(119903) (106)


gt



119877(x1015840) + U



11987110158401015840(x10158401015840) with

120588(B1198711015840(x1015840)B



H(119877)

B = H(119877)

B1198711015840(x1015840) otimes 1

(11989910158401015840)+ 1(119899

1015840)otimesH(119877)

B11987110158401015840

(x10158401015840) (107)


B(119873)


119873= 4119898119873


119871(x) =

B1198711015840(x1015840) times B

11987110158401015840(x10158401015840) with dist(x1015840 x10158401015840) gt (119862

1198732119873)119871 = 2119898119871

so that with 119877119896= (119862

1198732119873)119871

119896 one has

120598 (119877119896) lt eminus2119898119871119896 lt

1

2eminus119871120573

119896 (108)


119871119896(u) =

B(1198991015840)

119871119896(uJ)timesB

(11989910158401015840)



(a) dist(x1015840 x10158401015840) gt 119877119896= (119862

1198732119873)119871

119896

(b) B(1198991015840)

119871119896(u1015840) and B(119899

10158401015840)

119871119896(u10158401015840) are119898-Loc

(c) B(119873)

119871119896(u) is 119864-NR


119871119896(u) is (119864119898)-NS

Proof Set 119877119896= (119862

1198732119873)119871


above Denote

H(119877119896119873)

u119896 = H(119877119896)

B(119873)119871119896 (u)

G (119864) = (HB(119873)119871119896 (u)minus 119864)

minus1

(109)

and G(119877119896)(119864) = (H(119877119896119873)



H(119877119896 119873)

u119896 = HB(1198991015840)119871119896

(u1015840) otimes 1(11989910158401015840)+ 1(119899

1015840)otimesHB(119899

1015840)119871119896

(x10158401015840) (110)

with u = (u1015840 u10158401015840) isin Z119889119899

1015840

times Z11988911989910158401015840


119886119887= 120582

119886+120583

119887 where 120582

119886 = Σ(HB119871119896 (u



119887 = Σ(HB119871119896 (u


of H(119877119896 119873)


119886otimes 120595

119887 where


1015840)119871119896


of HB(11989910158401015840)119871119896


119886+ 120583

119887




|119864 minus (120582119886+ 120583

119887)| ge e119871

120573


119887)| ge e119871

120573

119896 and alsoas |(119864 minus 120583

119887) minus 120582

119886)| ge e119871

120573


G(119877119896) (u y 119864)

= sum

120582119886

sum

120583119887

120601119886(u1015840)120601

119886(y1015840)120595

119887(u10158401015840)120595

119887(y10158401015840)

(120582119886+ 120583

119887) minus 119864

= sum

120582119886

P1015840

119886(u1015840 y1015840)G(119877119896)

B119871119896 (u10158401015840)(u10158401015840 y10158401015840 119864 minus 120582

119886)

= sum

120583119887

P10158401015840

119887(u10158401015840 y10158401015840)G(119877119896)

B119871119896 (u1015840)(u1015840 y1015840 119864 minus 120583

119887)

(111)


B119871119896 (u1015840)(119864minus120583

119887) andG(119877119896)

B119871119896 (u10158401015840)(119864minus120582

119886) are

nonresonant1003817100381710038171003817100381710038171003817G(119877119896)

B119871119896 (u1015840)(119864 minus 120583

119887)

1003817100381710038171003817100381710038171003817le e119871

120573

119896

1003817100381710038171003817100381710038171003817G(119877119896)

B119871119896 (u10158401015840)(119864 minus 120582

119886)

1003817100381710038171003817100381710038171003817le e119871

120573

119896

(112)


y1015840) = 119871119896 in which case we

infer from (b)

10038161003816100381610038161003816G(119877119896) (u y 119864)10038161003816100381610038161003816 le

10038161003816100381610038161003816B119871119896(u10158401015840)10038161003816100381610038161003816eminus120574(119898119871119896119873minus1)119871119896+119871

120573

119896 (113)

or 120588(u10158401015840 y10158401015840) = 119871


10038161003816100381610038161003816G(119877119896) (u y 119864)10038161003816100381610038161003816 le

10038161003816100381610038161003816B119871119896(u1015840)10038161003816100381610038161003816eminus120574(119898119871119896119873minus1)119871119896+119871

120573

119896 (114)



119871119896+ 119871

120573

119896+ 119862 ln 119871

119896)

lt1

2eminus120574(119898119871119896119873)

(115)




10038171003817100381710038171003817G minus G(119877119896)

10038171003817100381710038171003817le10038171003817100381710038171003817U minus U(119877119896)

10038171003817100381710038171003817

10038171003817100381710038171003817G(119877119896)

10038171003817100381710038171003817G le eminus2119898119871119896e2119871

120573

119896

le1

2eminus120574(119898119871119896119873)

(117)


10038161003816100381610038161003816G (x y 119864) minus G(119877119896) (x y 119864)10038161003816100381610038161003816 le

1

2eminus120574(119898119871119896 119873)


119871119896+1(u)

(118)

Lemma 33 Let B119871119896+1(x)B



P B119871119896+1(x) B

119871119896+1(y) 119886119903119890 (119864119898) -119878

lt1

4119871minus119875(119873119896+1)

119896+1

(119)

Proof Let

R(2)= exist119864 B

119871119896+1(x) B

119871119896+1(y) 119886119903119890 119864-R

S(2)= B

119871119896+1(x) B119871119896+1

(y) 119886119903119890 (119864119898) -S (120)

ThenPS(2) le PR(2)

+PS(2)R(2)


119871119896+1(x) B

119871119896+1(y) must be 119864-NR


B(119873)

119871119896+1(x) = B(119899

1015840)

119871119896+1(x1015840) times B(119899

1015840)

119871119896+1(x10158401015840) (121)


119871119896+1(x10158401015840) are119898-Loc and

B119871119896+1(x) is (119864119898)-S then B




B(119873)

119871119896+1(y) = B(119899

1015840)

119871119896+1(y1015840) times B(119899

10158401015840)

119871119896+1(y10158401015840) (122)

If both B(1198991015840)119871119896+1(y1015840) and B(11989910158401015840)

119871119896+1(y10158401015840) are 119898-Loc and

B(119873)

119871119896+1(y) is (119864119898)-S then B(119873)

119871119896+1(y)must be 119864-R Let


1015840)

119871119896+1(x1015840) B(119899

10158401015840)

119871119896+1(x10158401015840)

B(1198991015840)

119871119896+1(y1015840) B(119899

1015840)

119871119896+1(y10158401015840) is 119898-NL119900119888

(123)


P L(4) le 4119871

minus119875(119873minus1119896+1)

119896+1= 4119871

minus2119875(119873119896+1)

119896+1 (124)



large enough

P S(2) le P R

(2) + P L

(4)

le eminus119871120573

119896 + 4119871minus2119875(119873119896+1)

119896+1lt1

4119871minus119875(119873119896+1)

119896+1

(125)






B119871119896(k) sub B

119871119896+1(u)with 120588(w k) ge 119862

1198731198711+120575

119896is (119864119898)-NS In the







119896also large)

1198711+984858

119896≫ 119871

1+120575


120588 (xB(119862119873minus1)119871119896

(w)) + 120588 (yB(119862119873minus1)119871119896(w))

ge 119877 minus (2119862119873minus 2) 119871

1+120575

119896

(126)

Assume first that

1199031015840= 120588 (xB

(119862119873minus1)1198711+120575119896(w)) ge 119871

119896+ 1

11990310158401015840= 120588 (yB

(119862119873minus1)1198711+120575119896(w)) ge 119871119896 + 1

(127)


1199031015840(x) and in B

11990310158401015840(y) are


1)1198711+120575

119896minus 2 ge 119877 minus 2119862

1198731198711+120575

119896










(119877minus21198621198731198711+120575119896 minus2119871119896)(119871119896+1)

sdot e119871120573

119896+1] = 119898(1 +1

2119871minus120591

119896)

119869119871119896

119871119896+ 1119877(1 minus

3119862119873

119877

sdot 119871119896) minus 119871

120573

119896+1= 119898119877[(1 +

1

2119871minus120591

119896)

119869


119896)


119873119871119896

1198711+984858

119896

) minus119871120573

119896+1

119898119877] = 119898119877[(1 +

1

2119871minus120591

119896)

119869


119896) (1 minus 3119862

119873119871minus984858

119896) minus

119871120573

119896+1

1198981198711+984858

119896

] ge 119898(1 +1

4


119896)

119869

119877 ge 120574 (119898 119871119896+1) 119877

(128)






1 119909

119873) 119909





1003816100381610038161003816 le 119862119889119871119889 (129)





by the functions

Φa =1

radic119873

sum

120587isinS119873

119873

⨂

119895=1

1119886120587minus1(119895) (130)

where a = 1198861 119886

119873 with 119886

1 119886

119873 = 119873



set is

ZNgt

= a = 1198861 119886

119873 119886

119895isinZ 119886

1 119886

119873 = 119873

(131)



1 119888

2 119888

119873 b = 119887

1 119888

2 119888

119873 with

1198861 119887

1 119888

2 119888

119873 = 119873 + 1

(ii) dZ(1198861 1198871) = 1


120588 (x y) = min120587isinS119873

max1le119895le119873

dZ (119909120587minus1(119895) 119910119895) (132)




119892V(x 120596) + U(x) where

U (x) = sum119894 =119895

119880(d (119909119894 119909

119895)) 119880 N 997888rarr R (133)


V (x 120596) = 119881 (1199091 120596) + sdot sdot sdot + 119881 (119909

119873 120596) (134)






Appendix



V (1199091 119909

119873 120596) = sum

119910isinxn119910119881 (119910 120596) (A1)





119908(y)) 119881 (119908 120596)

+ sum

V =119908

119888V119881 (V 120596) (A2)


119888V = nV (x) minus nV (y) (A3)



119908 while

n119908(x) minus n

119908(y) = 119888

119908isin Z 0 hence 1003816100381610038161003816119888119908

1003816100381610038161003816 ge 1 (A4)

Therefore

P 1003816100381610038161003816119892V (x 120596) minus 119892V (y 120596)

1003816100381610038161003816 le 119904

= E [P 1003816100381610038161003816V (x 120596) minus V (y 120596)

1003816100381610038161003816 le10038161003816100381610038161198921003816100381610038161003816

minus1119904 | F

=119908]

= E [P 1003816100381610038161003816119888119908119881 (119908 120596) | z (120596)

1003816100381610038161003816 le10038161003816100381610038161198921003816100381610038161003816

minus1119904 | F

=119908]

(A5)




120596 119888119908119881 (119908 120596) isin [z (120596) minus

10038161003816100381610038161198921003816100381610038161003816

minus1119904 z (120596) +

10038161003816100381610038161198921003816100381610038161003816

minus1119904]

= 120596 119881 (119908 120596)

isin [z (120596) minus11990410038161003816100381610038161198921003816100381610038161003816

minus1

119888119908

z (120596) +11990410038161003816100381610038161198921003816100381610038161003816

minus1

119888119908

]

(A6)


P 1003816100381610038161003816V (x 120596) minus V (y 120596)

1003816100381610038161003816 le10038161003816100381610038161198921003816100381610038161003816

minus1119904

le sup119886isinR

(119865119881119908(119886 + 2119904) minus 119865

119881119908(119886))

(A7)


at 119908 given F=119908 Since 119865




1003816100381610038161003816 le 119904 (A8)


1198710(u) admits



0 To prove the




100381610038161003816100381610038161003816GB1198710 (u)

(x y 119864)100381610038161003816100381610038161003816le eminus119898(119904)d(xy)

(A9)





119899

119895=11198612119871(119909

119895) If




1198711+120575(x) can be


(⋃

119895isinJ

1198612119871(119909

119895))∐(⋃

119894isinJ119888

1198612119871(119909

119894)) (A10)


120588 (119861119871(119909

119894) 119861

119871(119909

119895)) gt 2119871 (A11)


119873119871 then for some 119895

∘ 120588(119909

119895∘ 119910

119895∘) gt 119862

119873119871


120588 (119909119895 119909

119895∘) le diamΠx le 119860

119873119871

120588 (119910119895 119910

119895∘) le diamΠy lt 119860

119873119871

(A12)


120588 (119909119894 119910

119895) ge 120588 (119909

119895∘ 119910

119895∘) minus 120588 (119909

119894 119909

119895∘) minus 120588 (119910

119895 119910

119895∘)

ge 119862119873119871 minus 2119860

119873119871 ge 3119871

(A13)


yielding

120588 (Π119861119871(x) Π119861

119871(y)) gt 3119871 minus 2119871 = 119871 (A14)



Acknowledgments


References

































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119873

y = 1199101 119910

119873


dZ (119909119894 y) max119894

dZ (119910119894 x)] (31)

where

dZ (119909 y) = min119895

dZ (119909 119910119895) (32)


E[sup119905isinR

10038161003816100381610038161003816⟨1y10038161003816100381610038161003816eminusi119905H(120596)10038161003816100381610038161003816

1x⟩10038161003816100381610038161003816] le 119862 (x) eminus119886 ln

1+119888dH(xy) (33)



119871(119904) le 119862119871

119860119904119899 119862119860 isin

(0 +infin) and 119887 isin (0 1]





gt |Λ| equiv


HΛ (120596) = 1

ΛH (120596) 1Λ HΛ (34)







(Γxy119891) (x) = 120575xy119891 (y) (35)


Λ(119864) = (H

Λminus119864)




10038161003816100381610038161003816

le 119862ℓ


10038161003816100381610038161003816GBℓ(x) (x k)

10038161003816100381610038161003816max

k1015840isin120597+Bℓ(x)

10038161003816100381610038161003816GB119871 (k

1015840 y)10038161003816100381610038161003816

(36)

where

119862ℓ=1003816100381610038161003816120597Bℓ

(k)1003816100381610038161003816 le 119862 (119873) ℓ119873 (37)


10038161003816100381610038161003816G

119861119871(x y)10038161003816100381610038161003816

le (119862ℓmax

120588(xk)=ℓ

10038161003816100381610038161003816GBℓ(x) (x k)

10038161003816100381610038161003816) max

120588(uk1015840)leℓ+1

10038161003816100381610038161003816GB119871 (k

1015840 y)10038161003816100381610038161003816

(38)


10038161003816100381610038161003816GB119871 (x y 119864)

10038161003816100381610038161003816

le 119862ℓ

10038171003817100381710038171003817GBℓ(u) (119864)

10038171003817100381710038171003817max

k120588(uk)leℓ+1

10038161003816100381610038161003816GB119871 (k y 119864)

10038161003816100381610038161003816

(39)


ℓ(u)

1003816100381610038161003816120595 (x)1003816100381610038161003816 le 119862ℓ

10038171003817100381710038171003817GBℓ(x) (119864)

10038171003817100381710038171003817max

y120588(xy)leℓ+11003816100381610038161003816120595 (y)

1003816100381610038161003816 (40)



gt


119877(119906) rarr R



ℓ(119909) sub Λ

119877(119906) one has

119891 (119909) le 119902 max119910d(119909119910)leℓ

119891 (119910) (41)





M (119891 119861)

le 119902(119871minusℓ)(ℓ+1)

M (119891 119861)

M (119891 Λ) = max119909isinΛ

1003816100381610038161003816119891 (119909)1003816100381610038161003816

(42)





M (119891 119861)

(43)





1015840) ⊊ Λ and in 11990910158401015840 isin 119861

11990310158401015840(119906

10158401015840) ⊊ Λ with

d(1199061015840 11990610158401015840) ge 1199031015840 + 11990310158401015840 + 2 Then

119891 (1199061015840 119906

10158401015840) le 119902


10158401015840+1)(ℓ+1)rfloor

M (119891 119861)

le 119902(1199031015840+11990310158401015840minus2ℓ)(ℓ+1)

M (119891 119861)

(44)



tion

11989111991010158401015840 119910

1015840997891997888rarr 119891 (119910

1015840 119910

10158401015840) (45)


1015840) ⊊ Λ so by Lemma 6

11989111991010158401015840 (119906

1015840) = 119891 (119906

1015840 119910

10158401015840) le 119902

lfloor(1199031015840+1)(ℓ+1)rfloor

M (119891 Λ times Λ) (46)


10158401015840997891rarr 119891(119906

1015840 119910

10158401015840) which is

(ℓ 119902)-dominated in 11986111990310158401015840(119906






119896+1=

lceil119871120572

119896rceil 119871



0to be large enough


119871(u) is called




ℓ(u) sube B

119871(u) with ℓ ge 1198711120572





10038161003816100381610038161003816GB119871(u) (u y 119864 120596)

10038161003816100381610038161003816le eminus120574(119898119871)119871+2119871

120573

(47)

where

120574 (119898 119871) = 119898 (1 + 119871minus120591) 120591 =

1

8 (48)




120595119895 119895 = 1 |B



0isin B

119871119896(u)withd(x

0 y

0) gt 119871

119896minus1+1

and an integer

119877 isin [119871119896minus1 d (x

0 y

0) minus 119871

119896minus1+ 1] (49)


119877(x

0) is (119864119898)-NS and set

119902 = eminus120574(119898ℓ)ℓ (50)

Then



120595119895=

|120595119895⟩⟨120595





B119877(x

0) ni x 997891997888rarr

100381610038161003816100381610038161003816GB119871119896 (u)

(x y 119864)100381610038161003816100381610038161003816

(51)


0) and globally



119877(y


0fixed




satisfies10038161003816100381610038161003816120595

119895(x)120595

119895(y)10038161003816100381610038161003816 le eminus120574(119898119871)120588(xy) (52)




120588(x y) ge 119862119873119871 with 119862

119873= 2119860

119873+ 3 119860

119873= 4119873



in Section 43)



ℓ(y) and119898-







1003816100381610038161003816GB (x y 119864)1003816100381610038161003816 le sum

119864119895isinΣ(HB)

10038161003816100381610038161003816120595

119895(x)1003816100381610038161003816100381610038161003816100381610038161003816120595

119895(y)10038161003816100381610038161003816

10038161003816100381610038161003816119864 minus 119864

119895

10038161003816100381610038161003816

(53)


120573


1003816100381610038161003816GB (x y 119864)1003816100381610038161003816 le eminus120574(119898119871)119871+119871

120573+ln|B|

le eminus120574(119898119871)119871+2119871120573

(54)



119898(1 + 119871minus120591

119896) 119871

119896minus 2119871

120573

119896= 119898119871

119896+ 119898119871

1minus120591

119896minus 2119871

120573

119896

ge 119898(1 +1

2119871minus120591

119896) 119871

119896

(55)




(1)




(119864119898)-NS(B) if a ball B


(119864119898)-NS




119871119896(k) sub B

119871119896(u) with






120588 (xB(119862119873minus1)119871119896

(w)) + 120588 (yB(119862119873minus1)119871119896(w))

ge 119877 minus (2119862119873minus 2) 119871

119896

(56)

Assume first that

1199031015840= 120588 (xB

(119862119873minus1)119871119896(w)) ge 119871

119896+ 1

11990310158401015840= 120588 (yB

(119862119873minus1)119871119896(w)) ge 119871

119896+ 1

(57)



119896are auto-


Furthermore

1199031015840+ 119903

10158401015840ge 119877 minus 2 (119862

119873minus 1) 119871

119896minus 2 ge 119877 minus 2119862

119873119871119896 (58)










(119877minus2119862119873119871119896minus2119871119896)(119871119896+1)

e119871120573

119896+1]

= 119898(1 +1

2119871minus120591

119896)119871119896

119871119896+ 1119877 (1 minus 3C

119873119877minus1119871119896) minus 119871

120573

119896+1

= 119898119877[(1 +1

2119871minus120591

119896) (1 minus 119871

minus1

119896) (1 minus 3119862

119873119871minus984858

119896)

minus119871120573

119896+1

119898119877] ge 119898(1 +

1

4119871minus120591

119896)119877 ge 120574 (119898 119871

119896+1) 119877

(59)




B119871119896(x) B


119871119896+1(u) is 119864-


119871119896(y) must


the hypothesis






119895 (k1015840 k10158401015840) 997891rarr |120595

119895(k1015840)120595

119895(k10158401015840)| where

120595119895 119895 isin [1 |B





119895 = 1 so the


119871119896+1(u) and

assume 120588(x y) = 119877 ge 1198711+984858119896




minus ln 10038161003816100381610038161003816120595119895 (x)120595119895 (y)10038161003816100381610038161003816


(119877minus2119862119873119871119896minus2119871119896)(119871119896+1)

]

(60)






120588 (x y) ge 1198711+984858119896minus1

0 lt 120591 lt 120588

1 + 984858 lt 120572

120573 lt 1 minus 120591

(61)


120572 =4

3

120588 =1

6

120591 =1

8

120573 =1

2

(62)




119901 gt 0 there is 1198920= 119892

0(119898 119901 119871


|119892| ge 1198920 any ball B

1198710(119906) and any 119864 isin R

P B1198710(u) 119894119904 (119864119898) -119878 le 119871minus119901

0

P B1198710(u) 119894119904 119898-NL119900119888 le 119871

minus119901

0

(63)




1015840isin (0 1] 1198601015840


119871(u1015840) B(119873)



P dist (Σ119871u1015840 Σ119871u10158401015840) le 119904 le (3119871)

2119873ℎ119871 (2119904) (64)


HB119871(u10158401015840) and

ℎ119871 (119904) = 119862

1015840101584011987111986010158401015840

119904119887and1198871015840

119887 and 1198871015840equiv min [119887 1198871015840] (65)

and 11986010158401015840= 119860

10158401015840(119860



u10158401015840) gt 4119873119871

P exist119864 isin R B119871(u1015840) B

119871(u10158401015840) 119886119903119890 119864-119875119877 le eminus119871

1205731015840

(66)





119873= 4119873 and





H(119873)

B119871(u) = H(1198991015840)

B119871(u1015840)otimes 1(119899

10158401015840)+ 1(119899

1015840)otimesH(119899

10158401015840)

B119871(u10158401015840)(67)

with 1198991015840+11989910158401015840 = 119873 u = (u1015840 u10158401015840) u1015840isinZn1015840

gt u10158401015840isinZn10158401015840

gt while for




(x) ΠJ119888B119871(x)) gt 2119871

(B) If B119871(x) B


then 120588(ΠB119871(x) ΠB

119871(y)) ge 2119871






B(119873)

119871119896(u) = B(119899

1015840)

119871119896(u1015840) times B(119899

10158401015840)

119871119896(u10158401015840) (68)


If the balls B(1198991015840)

119871119896minus1(u1015840) and B(119899

10158401015840)


B(119873)

119871119896(u) is also119898-Loc



HB119871119896 (u)= HB(119899

1015840)119871119896

(u1015840) otimes 1(11989910158401015840)+ 1(119899

1015840)otimesHB(119899

10158401015840)119871119896

(u10158401015840) (69)



119895= 120601

1198951015840 otimes120595

11989510158401015840 where 120601


1015840)119871119896



10158401015840)119871119896

(u10158401015840) Now


119895 (x)Ψ119895 (y)10038161003816100381610038161003816

=100381610038161003816100381610038161206011198951015840 (x1015840)120601

11989510158401015840 (x10158401015840)120595

1198951015840 (y1015840)120595

11989510158401015840 (y10158401015840)10038161003816100381610038161003816

(70)


1015840)

119871119896(u1015840) and B(119899

10158401015840)

119871119896(u10158401015840)


119875 (119899 119896) = 2119873minus119899119901 (1 + 120579)

119896 1 le 119899 le 119873 119896 ge 0 (71)



119871119896(u)

P B(119899)

119871119896(u) 119894119904 119898-NL119900119888 le 119871

minus119875(119899119896)

119896 (72)


119871119896(u) one has

P B(119873)

119871119896(u) 119894119904 119898-NL119900119888 le 2119871

minus2119875(119873119896)

119896le1

4119871minus119875(119873119896)

119896 (73)




1015840)

119871119896(u1015840) B(119899

10158401015840)

119871119896(u10158401015840) is 119898-NLoc


P B(119873)

119871119896(u) is 119898-NL119900119888

le P B(11989910158401015840)

119871119896(u10158401015840) is 119898-NL119900119888

+ P B(11989910158401015840)

119871119896(u10158401015840) is 119898-NL119900119888

le 119871minus119875(1198991015840119896)

119896+ 119871

minus119875(11989910158401015840119896)

119896le 2119871

minus2119875(119873119896)

119896lt1

4119871minus119875(119873119896)

119896

(74)





1198961015840 with 1198961015840 le 119896


119871119896(u)

P B(119873)

119871 119894(u) 119894119904 119898-NL119900119888 le 119871

minus119875(119873119894)

119894 (75)


119871119896+1(x) one has

P B(119873)

119871119896+1(u) 119894119904 119898-NL119900119888 le 119871

minus119875(119873119896+1)

119896+1 (76)

provided that

119901 gt2120572

2

2 minus 1205722119873119889

0 lt 3120579 le min2 minus 1205722

1205722minus2119873119889


(77)


119871119896(u) B(119873)

119871119896(k) one

has for some 119886 119888 gt 0


119871119896(u) B(119873)

119871119896(k) 119886119903119890 (119864119898) -119878

le 119871minus119875(119873119896)

119896equiv 119871

minus119901(1+120599)119896


1+119888119871119896

(78)


N119896+1= B(119873)

119871119896+1(u) is 119898-NLo119888

S(2)

119896


119871119896(u) B(119873)

119871119896(k)

sub B(119873)

119871119896+1(w)

R(2)


119871119896(u) B(119873)

119871119896(k)

sub B(119873)

119871119896+1(w)

(79)


N119896+1sub S

(2)

119896subR

(2)

119896cup (S

(2)

119896R

(2)

119896) (80)


119896 le eminus119871

1205731015840


119896R

(2)


119871119896+1(w) and


S(2)

119896(u k) = exist119864 B(119873)

119871119896(u) B(119873)

119871119896(k) are (119864119898) -S (81)


119896(u k) R(2)

119896 either B(119873)

119871119896(u) or B(119873)

119871119896(k)


119871119896(u) is 119864-CNR



pair of 119898-NL119900119888 balls B(119873)

119871119896minus1(y1015840) B(119873)


(12)1198711+120575


(1) Both B(119873)

119871119896minus1(y1015840) and B(119873)

119871119896minus1(y10158401015840) are FI


120588 (ΠB119871119896minus1(u) ΠB

119871119896minus1(k)) ge 2119871

119896 (82)


119896minus1= 119871

11205722

119896+1and 119871

0large enough

P B(119873)

119871119896minus1(y1015840) B(119873)

119871119896minus1(y10158401015840) are 119898-NL119900119888

le 2119871minus(2119901120572

2)(1+120579)

119896minus1

119896+1

(83)


119871119896(w)|2 le (321198732)1198712119873


P S(2)

119896R

(2)

119896 le32119873

2119871minus(2119901120572

2)(1+120579)

119896minus1+2119873

119896+1 (84)



119896+1 under con-


2119901

12057222119873minus119873119901 (1 + 120579)

119896minus1minus 2119873119889 gt 2

119873minus119873119901 (1 + 120579)

119896+1 (85)

or equivalently

(1 + 120579)2lt2119901

1205722minus

2119873119889

119901 (1 + 120579)119896minus1 (86)


3120599 le2 minus 120572

2

1205722minus2119873

119901 (87)


P S(2)

119896 le P R

(2)

119896 + P S

(2)

119896R

(2)

119896

le eminus1198711205731015840

119896 +1

4119871minus119875(119873119896+1)

119896+1lt1

2119871minus119875(119873119896+1)

119896+1

(88)


119871119896minus1(y1015840) orB(119873)


P B(119873)

119871119896minus1(y1015840) B(119873)

119871119896minus1(y10158401015840) is 119898-NL119900119888

le min (P B(119873)

119871119896minus1(y1015840) is 119898-NL119900119888

P B(119873)

119871119896minus1(y10158401015840) is 119898-NL119900119888) le 119871

minus2119901(1+120579)119896minus1

119896minus1

lt 119871minus119901(1+120579)

119896+1

119896+1

(89)



119871minus119875(119873119896)

119896equiv 119871

minus119901(1+120599)119896


1+119888119871119896 (90)




P B(119873)

119871119896(u) 119894119904 119898-NL119900119888 le

1

4119871minus119901(1+120579)

119896

119896(91)


minus119901(1+120579)119896


119895of operator HB(119873)119871119896 (u)



119896one

has10038161003816100381610038161003816Ψ119895(x) Ψ






1003816100381610038161003816Ψ (y)1003816100381610038161003816 le eminus119898y

(93)








119871119896(y) sub

B119871119896+2(z) Fix 120596 isin Ω1015840


2le 1 implies Ψ

infinle 1 there is


119899isin B

119871119896minus1(0) then






119871119896+2(z) with 120588(y z) isin [119871

119896+1 119871

119896+2)


1+120575

119896) is (120582119898)-NS Note that



119877(y) with

119877 = 120588(y z)minus21198711+120575119896minus1 gt (12)119871

1+2120575



minusln 1003816100381610038161003816Ψ (y)

1003816100381610038161003816

120588 (y z)ge 119898 (1 + 119871

minus120591

119896) (1 minus

2119871119896+ 1

120588 (y z))

ge 119898(1 +1

2119871minus120591

119896) gt 119898

(95)









119871(119909) B


P exist 119864 isin 119868 B119871(x) B

119871(y) 119886119903119890 (119864119898) -S le 119891 (119871) (96)


119871(119909) cup B

119871(119910) one has

E[ sup120601isinB1(119868)

10038161003816100381610038161003816⟨1x1003816100381610038161003816120601 (HΛ (120596))

1003816100381610038161003816 1y⟩10038161003816100381610038161003816]

le 119862119871119889119890minus119898119871+ 119891 (119871)

(97)




Λ sup B119871(119909) cup B

119871(119910)


10038161003816100381610038161003816⟨1x1003816100381610038161003816120601 (HΛ (120596))

1003816100381610038161003816 1y⟩10038161003816100381610038161003816] le 119862119890

minus119886 ln1+119888120588(xy) (98)










same 119888 119886 gt 0 as in (98)

E[ sup120601isinC1(119868)

10038161003816100381610038161003816⟨1x1003816100381610038161003816120601 (H (120596))

1003816100381610038161003816 1y⟩10038161003816100381610038161003816] le 119862eminus119886 ln

1+119888120588(xy) (99)




int120601 (120582) 119889120583xyB120596 (120582)

= ⟨1x1003816100381610038161003816 120601 (HB (120596))Π119868

(HB (120596))100381610038161003816100381610038161y⟩

(100)








119899rarr H with







119871119896

E [1003816100381610038161003816120583

xy120596


E [100381610038161003816100381610038161003816120583xyB119871119896

100381610038161003816100381610038161003816(E)] (101)




120601119905 120582 997891997888rarr 119891

119868(120582) ei119905120582 119905 isin R (102)








984858 = 2120575

120572 = 1 + 4120575

120591 =120575

2

(103)



120574 (119898 119871 119899) = 119898 (1 + 119871minus120591)119873minus119899+1

(104)



120588(x y) ge 1198621198731198711+120575 with 119862

119873= 4119873






119871(x) admits

a factorization

B(119873)

119871(x) = B(119899

1015840)

119871(x1015840) times B(119899

10158401015840)

119871(x10158401015840) (105)


119871(x) B


then 120588(ΠB119871(x) ΠB

119871(y)) ge 1198711+120575




119877119877 ge 119903



119880(2)

119877(119903) = 1

119903le119877119880

(2)(119903) (106)


gt



119877(x1015840) + U



11987110158401015840(x10158401015840) with

120588(B1198711015840(x1015840)B



H(119877)

B = H(119877)

B1198711015840(x1015840) otimes 1

(11989910158401015840)+ 1(119899

1015840)otimesH(119877)

B11987110158401015840

(x10158401015840) (107)


B(119873)


119873= 4119898119873


119871(x) =

B1198711015840(x1015840) times B

11987110158401015840(x10158401015840) with dist(x1015840 x10158401015840) gt (119862

1198732119873)119871 = 2119898119871

so that with 119877119896= (119862

1198732119873)119871

119896 one has

120598 (119877119896) lt eminus2119898119871119896 lt

1

2eminus119871120573

119896 (108)


119871119896(u) =

B(1198991015840)

119871119896(uJ)timesB

(11989910158401015840)



(a) dist(x1015840 x10158401015840) gt 119877119896= (119862

1198732119873)119871

119896

(b) B(1198991015840)

119871119896(u1015840) and B(119899

10158401015840)

119871119896(u10158401015840) are119898-Loc

(c) B(119873)

119871119896(u) is 119864-NR


119871119896(u) is (119864119898)-NS

Proof Set 119877119896= (119862

1198732119873)119871


above Denote

H(119877119896119873)

u119896 = H(119877119896)

B(119873)119871119896 (u)

G (119864) = (HB(119873)119871119896 (u)minus 119864)

minus1

(109)

and G(119877119896)(119864) = (H(119877119896119873)



H(119877119896 119873)

u119896 = HB(1198991015840)119871119896

(u1015840) otimes 1(11989910158401015840)+ 1(119899

1015840)otimesHB(119899

1015840)119871119896

(x10158401015840) (110)

with u = (u1015840 u10158401015840) isin Z119889119899

1015840

times Z11988911989910158401015840


119886119887= 120582

119886+120583

119887 where 120582

119886 = Σ(HB119871119896 (u



119887 = Σ(HB119871119896 (u


of H(119877119896 119873)


119886otimes 120595

119887 where


1015840)119871119896


of HB(11989910158401015840)119871119896


119886+ 120583

119887




|119864 minus (120582119886+ 120583

119887)| ge e119871

120573


119887)| ge e119871

120573

119896 and alsoas |(119864 minus 120583

119887) minus 120582

119886)| ge e119871

120573


G(119877119896) (u y 119864)

= sum

120582119886

sum

120583119887

120601119886(u1015840)120601

119886(y1015840)120595

119887(u10158401015840)120595

119887(y10158401015840)

(120582119886+ 120583

119887) minus 119864

= sum

120582119886

P1015840

119886(u1015840 y1015840)G(119877119896)

B119871119896 (u10158401015840)(u10158401015840 y10158401015840 119864 minus 120582

119886)

= sum

120583119887

P10158401015840

119887(u10158401015840 y10158401015840)G(119877119896)

B119871119896 (u1015840)(u1015840 y1015840 119864 minus 120583

119887)

(111)


B119871119896 (u1015840)(119864minus120583

119887) andG(119877119896)

B119871119896 (u10158401015840)(119864minus120582

119886) are

nonresonant1003817100381710038171003817100381710038171003817G(119877119896)

B119871119896 (u1015840)(119864 minus 120583

119887)

1003817100381710038171003817100381710038171003817le e119871

120573

119896

1003817100381710038171003817100381710038171003817G(119877119896)

B119871119896 (u10158401015840)(119864 minus 120582

119886)

1003817100381710038171003817100381710038171003817le e119871

120573

119896

(112)


y1015840) = 119871119896 in which case we

infer from (b)

10038161003816100381610038161003816G(119877119896) (u y 119864)10038161003816100381610038161003816 le

10038161003816100381610038161003816B119871119896(u10158401015840)10038161003816100381610038161003816eminus120574(119898119871119896119873minus1)119871119896+119871

120573

119896 (113)

or 120588(u10158401015840 y10158401015840) = 119871


10038161003816100381610038161003816G(119877119896) (u y 119864)10038161003816100381610038161003816 le

10038161003816100381610038161003816B119871119896(u1015840)10038161003816100381610038161003816eminus120574(119898119871119896119873minus1)119871119896+119871

120573

119896 (114)



119871119896+ 119871

120573

119896+ 119862 ln 119871

119896)

lt1

2eminus120574(119898119871119896119873)

(115)




10038171003817100381710038171003817G minus G(119877119896)

10038171003817100381710038171003817le10038171003817100381710038171003817U minus U(119877119896)

10038171003817100381710038171003817

10038171003817100381710038171003817G(119877119896)

10038171003817100381710038171003817G le eminus2119898119871119896e2119871

120573

119896

le1

2eminus120574(119898119871119896119873)

(117)


10038161003816100381610038161003816G (x y 119864) minus G(119877119896) (x y 119864)10038161003816100381610038161003816 le

1

2eminus120574(119898119871119896 119873)


119871119896+1(u)

(118)

Lemma 33 Let B119871119896+1(x)B



P B119871119896+1(x) B

119871119896+1(y) 119886119903119890 (119864119898) -119878

lt1

4119871minus119875(119873119896+1)

119896+1

(119)

Proof Let

R(2)= exist119864 B

119871119896+1(x) B

119871119896+1(y) 119886119903119890 119864-R

S(2)= B

119871119896+1(x) B119871119896+1

(y) 119886119903119890 (119864119898) -S (120)

ThenPS(2) le PR(2)

+PS(2)R(2)


119871119896+1(x) B

119871119896+1(y) must be 119864-NR


B(119873)

119871119896+1(x) = B(119899

1015840)

119871119896+1(x1015840) times B(119899

1015840)

119871119896+1(x10158401015840) (121)


119871119896+1(x10158401015840) are119898-Loc and

B119871119896+1(x) is (119864119898)-S then B




B(119873)

119871119896+1(y) = B(119899

1015840)

119871119896+1(y1015840) times B(119899

10158401015840)

119871119896+1(y10158401015840) (122)

If both B(1198991015840)119871119896+1(y1015840) and B(11989910158401015840)

119871119896+1(y10158401015840) are 119898-Loc and

B(119873)

119871119896+1(y) is (119864119898)-S then B(119873)

119871119896+1(y)must be 119864-R Let


1015840)

119871119896+1(x1015840) B(119899

10158401015840)

119871119896+1(x10158401015840)

B(1198991015840)

119871119896+1(y1015840) B(119899

1015840)

119871119896+1(y10158401015840) is 119898-NL119900119888

(123)


P L(4) le 4119871

minus119875(119873minus1119896+1)

119896+1= 4119871

minus2119875(119873119896+1)

119896+1 (124)



large enough

P S(2) le P R

(2) + P L

(4)

le eminus119871120573

119896 + 4119871minus2119875(119873119896+1)

119896+1lt1

4119871minus119875(119873119896+1)

119896+1

(125)






B119871119896(k) sub B

119871119896+1(u)with 120588(w k) ge 119862

1198731198711+120575

119896is (119864119898)-NS In the







119896also large)

1198711+984858

119896≫ 119871

1+120575


120588 (xB(119862119873minus1)119871119896

(w)) + 120588 (yB(119862119873minus1)119871119896(w))

ge 119877 minus (2119862119873minus 2) 119871

1+120575

119896

(126)

Assume first that

1199031015840= 120588 (xB

(119862119873minus1)1198711+120575119896(w)) ge 119871

119896+ 1

11990310158401015840= 120588 (yB

(119862119873minus1)1198711+120575119896(w)) ge 119871119896 + 1

(127)


1199031015840(x) and in B

11990310158401015840(y) are


1)1198711+120575

119896minus 2 ge 119877 minus 2119862

1198731198711+120575

119896










(119877minus21198621198731198711+120575119896 minus2119871119896)(119871119896+1)

sdot e119871120573

119896+1] = 119898(1 +1

2119871minus120591

119896)

119869119871119896

119871119896+ 1119877(1 minus

3119862119873

119877

sdot 119871119896) minus 119871

120573

119896+1= 119898119877[(1 +

1

2119871minus120591

119896)

119869


119896)


119873119871119896

1198711+984858

119896

) minus119871120573

119896+1

119898119877] = 119898119877[(1 +

1

2119871minus120591

119896)

119869


119896) (1 minus 3119862

119873119871minus984858

119896) minus

119871120573

119896+1

1198981198711+984858

119896

] ge 119898(1 +1

4


119896)

119869

119877 ge 120574 (119898 119871119896+1) 119877

(128)






1 119909

119873) 119909





1003816100381610038161003816 le 119862119889119871119889 (129)





by the functions

Φa =1

radic119873

sum

120587isinS119873

119873

⨂

119895=1

1119886120587minus1(119895) (130)

where a = 1198861 119886

119873 with 119886

1 119886

119873 = 119873



set is

ZNgt

= a = 1198861 119886

119873 119886

119895isinZ 119886

1 119886

119873 = 119873

(131)



1 119888

2 119888

119873 b = 119887

1 119888

2 119888

119873 with

1198861 119887

1 119888

2 119888

119873 = 119873 + 1

(ii) dZ(1198861 1198871) = 1


120588 (x y) = min120587isinS119873

max1le119895le119873

dZ (119909120587minus1(119895) 119910119895) (132)




119892V(x 120596) + U(x) where

U (x) = sum119894 =119895

119880(d (119909119894 119909

119895)) 119880 N 997888rarr R (133)


V (x 120596) = 119881 (1199091 120596) + sdot sdot sdot + 119881 (119909

119873 120596) (134)






Appendix



V (1199091 119909

119873 120596) = sum

119910isinxn119910119881 (119910 120596) (A1)





119908(y)) 119881 (119908 120596)

+ sum

V =119908

119888V119881 (V 120596) (A2)


119888V = nV (x) minus nV (y) (A3)



119908 while

n119908(x) minus n

119908(y) = 119888

119908isin Z 0 hence 1003816100381610038161003816119888119908

1003816100381610038161003816 ge 1 (A4)

Therefore

P 1003816100381610038161003816119892V (x 120596) minus 119892V (y 120596)

1003816100381610038161003816 le 119904

= E [P 1003816100381610038161003816V (x 120596) minus V (y 120596)

1003816100381610038161003816 le10038161003816100381610038161198921003816100381610038161003816

minus1119904 | F

=119908]

= E [P 1003816100381610038161003816119888119908119881 (119908 120596) | z (120596)

1003816100381610038161003816 le10038161003816100381610038161198921003816100381610038161003816

minus1119904 | F

=119908]

(A5)




120596 119888119908119881 (119908 120596) isin [z (120596) minus

10038161003816100381610038161198921003816100381610038161003816

minus1119904 z (120596) +

10038161003816100381610038161198921003816100381610038161003816

minus1119904]

= 120596 119881 (119908 120596)

isin [z (120596) minus11990410038161003816100381610038161198921003816100381610038161003816

minus1

119888119908

z (120596) +11990410038161003816100381610038161198921003816100381610038161003816

minus1

119888119908

]

(A6)


P 1003816100381610038161003816V (x 120596) minus V (y 120596)

1003816100381610038161003816 le10038161003816100381610038161198921003816100381610038161003816

minus1119904

le sup119886isinR

(119865119881119908(119886 + 2119904) minus 119865

119881119908(119886))

(A7)


at 119908 given F=119908 Since 119865




1003816100381610038161003816 le 119904 (A8)


1198710(u) admits



0 To prove the




100381610038161003816100381610038161003816GB1198710 (u)

(x y 119864)100381610038161003816100381610038161003816le eminus119898(119904)d(xy)

(A9)





119899

119895=11198612119871(119909

119895) If




1198711+120575(x) can be


(⋃

119895isinJ

1198612119871(119909

119895))∐(⋃

119894isinJ119888

1198612119871(119909

119894)) (A10)


120588 (119861119871(119909

119894) 119861

119871(119909

119895)) gt 2119871 (A11)


119873119871 then for some 119895

∘ 120588(119909

119895∘ 119910

119895∘) gt 119862

119873119871


120588 (119909119895 119909

119895∘) le diamΠx le 119860

119873119871

120588 (119910119895 119910

119895∘) le diamΠy lt 119860

119873119871

(A12)


120588 (119909119894 119910

119895) ge 120588 (119909

119895∘ 119910

119895∘) minus 120588 (119909

119894 119909

119895∘) minus 120588 (119910

119895 119910

119895∘)

ge 119862119873119871 minus 2119860

119873119871 ge 3119871

(A13)


yielding

120588 (Π119861119871(x) Π119861

119871(y)) gt 3119871 minus 2119871 = 119871 (A14)



Acknowledgments


References

































Volume 2014




Journal of











Journal of


Function Spaces






Algebra













1015840) ⊊ Λ and in 11990910158401015840 isin 119861

11990310158401015840(119906

10158401015840) ⊊ Λ with

d(1199061015840 11990610158401015840) ge 1199031015840 + 11990310158401015840 + 2 Then

119891 (1199061015840 119906

10158401015840) le 119902


10158401015840+1)(ℓ+1)rfloor

M (119891 119861)

le 119902(1199031015840+11990310158401015840minus2ℓ)(ℓ+1)

M (119891 119861)

(44)



tion

11989111991010158401015840 119910

1015840997891997888rarr 119891 (119910

1015840 119910

10158401015840) (45)


1015840) ⊊ Λ so by Lemma 6

11989111991010158401015840 (119906

1015840) = 119891 (119906

1015840 119910

10158401015840) le 119902

lfloor(1199031015840+1)(ℓ+1)rfloor

M (119891 Λ times Λ) (46)


10158401015840997891rarr 119891(119906

1015840 119910

10158401015840) which is

(ℓ 119902)-dominated in 11986111990310158401015840(119906






119896+1=

lceil119871120572

119896rceil 119871



0to be large enough


119871(u) is called




ℓ(u) sube B

119871(u) with ℓ ge 1198711120572





10038161003816100381610038161003816GB119871(u) (u y 119864 120596)

10038161003816100381610038161003816le eminus120574(119898119871)119871+2119871

120573

(47)

where

120574 (119898 119871) = 119898 (1 + 119871minus120591) 120591 =

1

8 (48)




120595119895 119895 = 1 |B



0isin B

119871119896(u)withd(x

0 y

0) gt 119871

119896minus1+1

and an integer

119877 isin [119871119896minus1 d (x

0 y

0) minus 119871

119896minus1+ 1] (49)


119877(x

0) is (119864119898)-NS and set

119902 = eminus120574(119898ℓ)ℓ (50)

Then



120595119895=

|120595119895⟩⟨120595





B119877(x

0) ni x 997891997888rarr

100381610038161003816100381610038161003816GB119871119896 (u)

(x y 119864)100381610038161003816100381610038161003816

(51)


0) and globally



119877(y


0fixed




satisfies10038161003816100381610038161003816120595

119895(x)120595

119895(y)10038161003816100381610038161003816 le eminus120574(119898119871)120588(xy) (52)




120588(x y) ge 119862119873119871 with 119862

119873= 2119860

119873+ 3 119860

119873= 4119873



in Section 43)



ℓ(y) and119898-







1003816100381610038161003816GB (x y 119864)1003816100381610038161003816 le sum

119864119895isinΣ(HB)

10038161003816100381610038161003816120595

119895(x)1003816100381610038161003816100381610038161003816100381610038161003816120595

119895(y)10038161003816100381610038161003816

10038161003816100381610038161003816119864 minus 119864

119895

10038161003816100381610038161003816

(53)


120573


1003816100381610038161003816GB (x y 119864)1003816100381610038161003816 le eminus120574(119898119871)119871+119871

120573+ln|B|

le eminus120574(119898119871)119871+2119871120573

(54)



119898(1 + 119871minus120591

119896) 119871

119896minus 2119871

120573

119896= 119898119871

119896+ 119898119871

1minus120591

119896minus 2119871

120573

119896

ge 119898(1 +1

2119871minus120591

119896) 119871

119896

(55)




(1)




(119864119898)-NS(B) if a ball B


(119864119898)-NS




119871119896(k) sub B

119871119896(u) with






120588 (xB(119862119873minus1)119871119896

(w)) + 120588 (yB(119862119873minus1)119871119896(w))

ge 119877 minus (2119862119873minus 2) 119871

119896

(56)

Assume first that

1199031015840= 120588 (xB

(119862119873minus1)119871119896(w)) ge 119871

119896+ 1

11990310158401015840= 120588 (yB

(119862119873minus1)119871119896(w)) ge 119871

119896+ 1

(57)



119896are auto-


Furthermore

1199031015840+ 119903

10158401015840ge 119877 minus 2 (119862

119873minus 1) 119871

119896minus 2 ge 119877 minus 2119862

119873119871119896 (58)










(119877minus2119862119873119871119896minus2119871119896)(119871119896+1)

e119871120573

119896+1]

= 119898(1 +1

2119871minus120591

119896)119871119896

119871119896+ 1119877 (1 minus 3C

119873119877minus1119871119896) minus 119871

120573

119896+1

= 119898119877[(1 +1

2119871minus120591

119896) (1 minus 119871

minus1

119896) (1 minus 3119862

119873119871minus984858

119896)

minus119871120573

119896+1

119898119877] ge 119898(1 +

1

4119871minus120591

119896)119877 ge 120574 (119898 119871

119896+1) 119877

(59)




B119871119896(x) B


119871119896+1(u) is 119864-


119871119896(y) must


the hypothesis






119895 (k1015840 k10158401015840) 997891rarr |120595

119895(k1015840)120595

119895(k10158401015840)| where

120595119895 119895 isin [1 |B





119895 = 1 so the


119871119896+1(u) and

assume 120588(x y) = 119877 ge 1198711+984858119896




minus ln 10038161003816100381610038161003816120595119895 (x)120595119895 (y)10038161003816100381610038161003816


(119877minus2119862119873119871119896minus2119871119896)(119871119896+1)

]

(60)






120588 (x y) ge 1198711+984858119896minus1

0 lt 120591 lt 120588

1 + 984858 lt 120572

120573 lt 1 minus 120591

(61)


120572 =4

3

120588 =1

6

120591 =1

8

120573 =1

2

(62)




119901 gt 0 there is 1198920= 119892

0(119898 119901 119871


|119892| ge 1198920 any ball B

1198710(119906) and any 119864 isin R

P B1198710(u) 119894119904 (119864119898) -119878 le 119871minus119901

0

P B1198710(u) 119894119904 119898-NL119900119888 le 119871

minus119901

0

(63)




1015840isin (0 1] 1198601015840


119871(u1015840) B(119873)



P dist (Σ119871u1015840 Σ119871u10158401015840) le 119904 le (3119871)

2119873ℎ119871 (2119904) (64)


HB119871(u10158401015840) and

ℎ119871 (119904) = 119862

1015840101584011987111986010158401015840

119904119887and1198871015840

119887 and 1198871015840equiv min [119887 1198871015840] (65)

and 11986010158401015840= 119860

10158401015840(119860



u10158401015840) gt 4119873119871

P exist119864 isin R B119871(u1015840) B

119871(u10158401015840) 119886119903119890 119864-119875119877 le eminus119871

1205731015840

(66)





119873= 4119873 and





H(119873)

B119871(u) = H(1198991015840)

B119871(u1015840)otimes 1(119899

10158401015840)+ 1(119899

1015840)otimesH(119899

10158401015840)

B119871(u10158401015840)(67)

with 1198991015840+11989910158401015840 = 119873 u = (u1015840 u10158401015840) u1015840isinZn1015840

gt u10158401015840isinZn10158401015840

gt while for




(x) ΠJ119888B119871(x)) gt 2119871

(B) If B119871(x) B


then 120588(ΠB119871(x) ΠB

119871(y)) ge 2119871






B(119873)

119871119896(u) = B(119899

1015840)

119871119896(u1015840) times B(119899

10158401015840)

119871119896(u10158401015840) (68)


If the balls B(1198991015840)

119871119896minus1(u1015840) and B(119899

10158401015840)


B(119873)

119871119896(u) is also119898-Loc



HB119871119896 (u)= HB(119899

1015840)119871119896

(u1015840) otimes 1(11989910158401015840)+ 1(119899

1015840)otimesHB(119899

10158401015840)119871119896

(u10158401015840) (69)



119895= 120601

1198951015840 otimes120595

11989510158401015840 where 120601


1015840)119871119896



10158401015840)119871119896

(u10158401015840) Now


119895 (x)Ψ119895 (y)10038161003816100381610038161003816

=100381610038161003816100381610038161206011198951015840 (x1015840)120601

11989510158401015840 (x10158401015840)120595

1198951015840 (y1015840)120595

11989510158401015840 (y10158401015840)10038161003816100381610038161003816

(70)


1015840)

119871119896(u1015840) and B(119899

10158401015840)

119871119896(u10158401015840)


119875 (119899 119896) = 2119873minus119899119901 (1 + 120579)

119896 1 le 119899 le 119873 119896 ge 0 (71)



119871119896(u)

P B(119899)

119871119896(u) 119894119904 119898-NL119900119888 le 119871

minus119875(119899119896)

119896 (72)


119871119896(u) one has

P B(119873)

119871119896(u) 119894119904 119898-NL119900119888 le 2119871

minus2119875(119873119896)

119896le1

4119871minus119875(119873119896)

119896 (73)




1015840)

119871119896(u1015840) B(119899

10158401015840)

119871119896(u10158401015840) is 119898-NLoc


P B(119873)

119871119896(u) is 119898-NL119900119888

le P B(11989910158401015840)

119871119896(u10158401015840) is 119898-NL119900119888

+ P B(11989910158401015840)

119871119896(u10158401015840) is 119898-NL119900119888

le 119871minus119875(1198991015840119896)

119896+ 119871

minus119875(11989910158401015840119896)

119896le 2119871

minus2119875(119873119896)

119896lt1

4119871minus119875(119873119896)

119896

(74)





1198961015840 with 1198961015840 le 119896


119871119896(u)

P B(119873)

119871 119894(u) 119894119904 119898-NL119900119888 le 119871

minus119875(119873119894)

119894 (75)


119871119896+1(x) one has

P B(119873)

119871119896+1(u) 119894119904 119898-NL119900119888 le 119871

minus119875(119873119896+1)

119896+1 (76)

provided that

119901 gt2120572

2

2 minus 1205722119873119889

0 lt 3120579 le min2 minus 1205722

1205722minus2119873119889


(77)


119871119896(u) B(119873)

119871119896(k) one

has for some 119886 119888 gt 0


119871119896(u) B(119873)

119871119896(k) 119886119903119890 (119864119898) -119878

le 119871minus119875(119873119896)

119896equiv 119871

minus119901(1+120599)119896


1+119888119871119896

(78)


N119896+1= B(119873)

119871119896+1(u) is 119898-NLo119888

S(2)

119896


119871119896(u) B(119873)

119871119896(k)

sub B(119873)

119871119896+1(w)

R(2)


119871119896(u) B(119873)

119871119896(k)

sub B(119873)

119871119896+1(w)

(79)


N119896+1sub S

(2)

119896subR

(2)

119896cup (S

(2)

119896R

(2)

119896) (80)


119896 le eminus119871

1205731015840


119896R

(2)


119871119896+1(w) and


S(2)

119896(u k) = exist119864 B(119873)

119871119896(u) B(119873)

119871119896(k) are (119864119898) -S (81)


119896(u k) R(2)

119896 either B(119873)

119871119896(u) or B(119873)

119871119896(k)


119871119896(u) is 119864-CNR



pair of 119898-NL119900119888 balls B(119873)

119871119896minus1(y1015840) B(119873)


(12)1198711+120575


(1) Both B(119873)

119871119896minus1(y1015840) and B(119873)

119871119896minus1(y10158401015840) are FI


120588 (ΠB119871119896minus1(u) ΠB

119871119896minus1(k)) ge 2119871

119896 (82)


119896minus1= 119871

11205722

119896+1and 119871

0large enough

P B(119873)

119871119896minus1(y1015840) B(119873)

119871119896minus1(y10158401015840) are 119898-NL119900119888

le 2119871minus(2119901120572

2)(1+120579)

119896minus1

119896+1

(83)


119871119896(w)|2 le (321198732)1198712119873


P S(2)

119896R

(2)

119896 le32119873

2119871minus(2119901120572

2)(1+120579)

119896minus1+2119873

119896+1 (84)



119896+1 under con-


2119901

12057222119873minus119873119901 (1 + 120579)

119896minus1minus 2119873119889 gt 2

119873minus119873119901 (1 + 120579)

119896+1 (85)

or equivalently

(1 + 120579)2lt2119901

1205722minus

2119873119889

119901 (1 + 120579)119896minus1 (86)


3120599 le2 minus 120572

2

1205722minus2119873

119901 (87)


P S(2)

119896 le P R

(2)

119896 + P S

(2)

119896R

(2)

119896

le eminus1198711205731015840

119896 +1

4119871minus119875(119873119896+1)

119896+1lt1

2119871minus119875(119873119896+1)

119896+1

(88)


119871119896minus1(y1015840) orB(119873)


P B(119873)

119871119896minus1(y1015840) B(119873)

119871119896minus1(y10158401015840) is 119898-NL119900119888

le min (P B(119873)

119871119896minus1(y1015840) is 119898-NL119900119888

P B(119873)

119871119896minus1(y10158401015840) is 119898-NL119900119888) le 119871

minus2119901(1+120579)119896minus1

119896minus1

lt 119871minus119901(1+120579)

119896+1

119896+1

(89)



119871minus119875(119873119896)

119896equiv 119871

minus119901(1+120599)119896


1+119888119871119896 (90)




P B(119873)

119871119896(u) 119894119904 119898-NL119900119888 le

1

4119871minus119901(1+120579)

119896

119896(91)


minus119901(1+120579)119896


119895of operator HB(119873)119871119896 (u)



119896one

has10038161003816100381610038161003816Ψ119895(x) Ψ






1003816100381610038161003816Ψ (y)1003816100381610038161003816 le eminus119898y

(93)








119871119896(y) sub

B119871119896+2(z) Fix 120596 isin Ω1015840


2le 1 implies Ψ

infinle 1 there is


119899isin B

119871119896minus1(0) then






119871119896+2(z) with 120588(y z) isin [119871

119896+1 119871

119896+2)


1+120575

119896) is (120582119898)-NS Note that



119877(y) with

119877 = 120588(y z)minus21198711+120575119896minus1 gt (12)119871

1+2120575



minusln 1003816100381610038161003816Ψ (y)

1003816100381610038161003816

120588 (y z)ge 119898 (1 + 119871

minus120591

119896) (1 minus

2119871119896+ 1

120588 (y z))

ge 119898(1 +1

2119871minus120591

119896) gt 119898

(95)









119871(119909) B


P exist 119864 isin 119868 B119871(x) B

119871(y) 119886119903119890 (119864119898) -S le 119891 (119871) (96)


119871(119909) cup B

119871(119910) one has

E[ sup120601isinB1(119868)

10038161003816100381610038161003816⟨1x1003816100381610038161003816120601 (HΛ (120596))

1003816100381610038161003816 1y⟩10038161003816100381610038161003816]

le 119862119871119889119890minus119898119871+ 119891 (119871)

(97)




Λ sup B119871(119909) cup B

119871(119910)


10038161003816100381610038161003816⟨1x1003816100381610038161003816120601 (HΛ (120596))

1003816100381610038161003816 1y⟩10038161003816100381610038161003816] le 119862119890

minus119886 ln1+119888120588(xy) (98)










same 119888 119886 gt 0 as in (98)

E[ sup120601isinC1(119868)

10038161003816100381610038161003816⟨1x1003816100381610038161003816120601 (H (120596))

1003816100381610038161003816 1y⟩10038161003816100381610038161003816] le 119862eminus119886 ln

1+119888120588(xy) (99)




int120601 (120582) 119889120583xyB120596 (120582)

= ⟨1x1003816100381610038161003816 120601 (HB (120596))Π119868

(HB (120596))100381610038161003816100381610038161y⟩

(100)








119899rarr H with







119871119896

E [1003816100381610038161003816120583

xy120596


E [100381610038161003816100381610038161003816120583xyB119871119896

100381610038161003816100381610038161003816(E)] (101)




120601119905 120582 997891997888rarr 119891

119868(120582) ei119905120582 119905 isin R (102)








984858 = 2120575

120572 = 1 + 4120575

120591 =120575

2

(103)



120574 (119898 119871 119899) = 119898 (1 + 119871minus120591)119873minus119899+1

(104)



120588(x y) ge 1198621198731198711+120575 with 119862

119873= 4119873






119871(x) admits

a factorization

B(119873)

119871(x) = B(119899

1015840)

119871(x1015840) times B(119899

10158401015840)

119871(x10158401015840) (105)


119871(x) B


then 120588(ΠB119871(x) ΠB

119871(y)) ge 1198711+120575




119877119877 ge 119903



119880(2)

119877(119903) = 1

119903le119877119880

(2)(119903) (106)


gt



119877(x1015840) + U



11987110158401015840(x10158401015840) with

120588(B1198711015840(x1015840)B



H(119877)

B = H(119877)

B1198711015840(x1015840) otimes 1

(11989910158401015840)+ 1(119899

1015840)otimesH(119877)

B11987110158401015840

(x10158401015840) (107)


B(119873)


119873= 4119898119873


119871(x) =

B1198711015840(x1015840) times B

11987110158401015840(x10158401015840) with dist(x1015840 x10158401015840) gt (119862

1198732119873)119871 = 2119898119871

so that with 119877119896= (119862

1198732119873)119871

119896 one has

120598 (119877119896) lt eminus2119898119871119896 lt

1

2eminus119871120573

119896 (108)


119871119896(u) =

B(1198991015840)

119871119896(uJ)timesB

(11989910158401015840)



(a) dist(x1015840 x10158401015840) gt 119877119896= (119862

1198732119873)119871

119896

(b) B(1198991015840)

119871119896(u1015840) and B(119899

10158401015840)

119871119896(u10158401015840) are119898-Loc

(c) B(119873)

119871119896(u) is 119864-NR


119871119896(u) is (119864119898)-NS

Proof Set 119877119896= (119862

1198732119873)119871


above Denote

H(119877119896119873)

u119896 = H(119877119896)

B(119873)119871119896 (u)

G (119864) = (HB(119873)119871119896 (u)minus 119864)

minus1

(109)

and G(119877119896)(119864) = (H(119877119896119873)



H(119877119896 119873)

u119896 = HB(1198991015840)119871119896

(u1015840) otimes 1(11989910158401015840)+ 1(119899

1015840)otimesHB(119899

1015840)119871119896

(x10158401015840) (110)

with u = (u1015840 u10158401015840) isin Z119889119899

1015840

times Z11988911989910158401015840


119886119887= 120582

119886+120583

119887 where 120582

119886 = Σ(HB119871119896 (u



119887 = Σ(HB119871119896 (u


of H(119877119896 119873)


119886otimes 120595

119887 where


1015840)119871119896


of HB(11989910158401015840)119871119896


119886+ 120583

119887




|119864 minus (120582119886+ 120583

119887)| ge e119871

120573


119887)| ge e119871

120573

119896 and alsoas |(119864 minus 120583

119887) minus 120582

119886)| ge e119871

120573


G(119877119896) (u y 119864)

= sum

120582119886

sum

120583119887

120601119886(u1015840)120601

119886(y1015840)120595

119887(u10158401015840)120595

119887(y10158401015840)

(120582119886+ 120583

119887) minus 119864

= sum

120582119886

P1015840

119886(u1015840 y1015840)G(119877119896)

B119871119896 (u10158401015840)(u10158401015840 y10158401015840 119864 minus 120582

119886)

= sum

120583119887

P10158401015840

119887(u10158401015840 y10158401015840)G(119877119896)

B119871119896 (u1015840)(u1015840 y1015840 119864 minus 120583

119887)

(111)


B119871119896 (u1015840)(119864minus120583

119887) andG(119877119896)

B119871119896 (u10158401015840)(119864minus120582

119886) are

nonresonant1003817100381710038171003817100381710038171003817G(119877119896)

B119871119896 (u1015840)(119864 minus 120583

119887)

1003817100381710038171003817100381710038171003817le e119871

120573

119896

1003817100381710038171003817100381710038171003817G(119877119896)

B119871119896 (u10158401015840)(119864 minus 120582

119886)

1003817100381710038171003817100381710038171003817le e119871

120573

119896

(112)


y1015840) = 119871119896 in which case we

infer from (b)

10038161003816100381610038161003816G(119877119896) (u y 119864)10038161003816100381610038161003816 le

10038161003816100381610038161003816B119871119896(u10158401015840)10038161003816100381610038161003816eminus120574(119898119871119896119873minus1)119871119896+119871

120573

119896 (113)

or 120588(u10158401015840 y10158401015840) = 119871


10038161003816100381610038161003816G(119877119896) (u y 119864)10038161003816100381610038161003816 le

10038161003816100381610038161003816B119871119896(u1015840)10038161003816100381610038161003816eminus120574(119898119871119896119873minus1)119871119896+119871

120573

119896 (114)



119871119896+ 119871

120573

119896+ 119862 ln 119871

119896)

lt1

2eminus120574(119898119871119896119873)

(115)




10038171003817100381710038171003817G minus G(119877119896)

10038171003817100381710038171003817le10038171003817100381710038171003817U minus U(119877119896)

10038171003817100381710038171003817

10038171003817100381710038171003817G(119877119896)

10038171003817100381710038171003817G le eminus2119898119871119896e2119871

120573

119896

le1

2eminus120574(119898119871119896119873)

(117)


10038161003816100381610038161003816G (x y 119864) minus G(119877119896) (x y 119864)10038161003816100381610038161003816 le

1

2eminus120574(119898119871119896 119873)


119871119896+1(u)

(118)

Lemma 33 Let B119871119896+1(x)B



P B119871119896+1(x) B

119871119896+1(y) 119886119903119890 (119864119898) -119878

lt1

4119871minus119875(119873119896+1)

119896+1

(119)

Proof Let

R(2)= exist119864 B

119871119896+1(x) B

119871119896+1(y) 119886119903119890 119864-R

S(2)= B

119871119896+1(x) B119871119896+1

(y) 119886119903119890 (119864119898) -S (120)

ThenPS(2) le PR(2)

+PS(2)R(2)


119871119896+1(x) B

119871119896+1(y) must be 119864-NR


B(119873)

119871119896+1(x) = B(119899

1015840)

119871119896+1(x1015840) times B(119899

1015840)

119871119896+1(x10158401015840) (121)


119871119896+1(x10158401015840) are119898-Loc and

B119871119896+1(x) is (119864119898)-S then B




B(119873)

119871119896+1(y) = B(119899

1015840)

119871119896+1(y1015840) times B(119899

10158401015840)

119871119896+1(y10158401015840) (122)

If both B(1198991015840)119871119896+1(y1015840) and B(11989910158401015840)

119871119896+1(y10158401015840) are 119898-Loc and

B(119873)

119871119896+1(y) is (119864119898)-S then B(119873)

119871119896+1(y)must be 119864-R Let


1015840)

119871119896+1(x1015840) B(119899

10158401015840)

119871119896+1(x10158401015840)

B(1198991015840)

119871119896+1(y1015840) B(119899

1015840)

119871119896+1(y10158401015840) is 119898-NL119900119888

(123)


P L(4) le 4119871

minus119875(119873minus1119896+1)

119896+1= 4119871

minus2119875(119873119896+1)

119896+1 (124)



large enough

P S(2) le P R

(2) + P L

(4)

le eminus119871120573

119896 + 4119871minus2119875(119873119896+1)

119896+1lt1

4119871minus119875(119873119896+1)

119896+1

(125)






B119871119896(k) sub B

119871119896+1(u)with 120588(w k) ge 119862

1198731198711+120575

119896is (119864119898)-NS In the







119896also large)

1198711+984858

119896≫ 119871

1+120575


120588 (xB(119862119873minus1)119871119896

(w)) + 120588 (yB(119862119873minus1)119871119896(w))

ge 119877 minus (2119862119873minus 2) 119871

1+120575

119896

(126)

Assume first that

1199031015840= 120588 (xB

(119862119873minus1)1198711+120575119896(w)) ge 119871

119896+ 1

11990310158401015840= 120588 (yB

(119862119873minus1)1198711+120575119896(w)) ge 119871119896 + 1

(127)


1199031015840(x) and in B

11990310158401015840(y) are


1)1198711+120575

119896minus 2 ge 119877 minus 2119862

1198731198711+120575

119896










(119877minus21198621198731198711+120575119896 minus2119871119896)(119871119896+1)

sdot e119871120573

119896+1] = 119898(1 +1

2119871minus120591

119896)

119869119871119896

119871119896+ 1119877(1 minus

3119862119873

119877

sdot 119871119896) minus 119871

120573

119896+1= 119898119877[(1 +

1

2119871minus120591

119896)

119869


119896)


119873119871119896

1198711+984858

119896

) minus119871120573

119896+1

119898119877] = 119898119877[(1 +

1

2119871minus120591

119896)

119869


119896) (1 minus 3119862

119873119871minus984858

119896) minus

119871120573

119896+1

1198981198711+984858

119896

] ge 119898(1 +1

4


119896)

119869

119877 ge 120574 (119898 119871119896+1) 119877

(128)






1 119909

119873) 119909





1003816100381610038161003816 le 119862119889119871119889 (129)





by the functions

Φa =1

radic119873

sum

120587isinS119873

119873

⨂

119895=1

1119886120587minus1(119895) (130)

where a = 1198861 119886

119873 with 119886

1 119886

119873 = 119873



set is

ZNgt

= a = 1198861 119886

119873 119886

119895isinZ 119886

1 119886

119873 = 119873

(131)



1 119888

2 119888

119873 b = 119887

1 119888

2 119888

119873 with

1198861 119887

1 119888

2 119888

119873 = 119873 + 1

(ii) dZ(1198861 1198871) = 1


120588 (x y) = min120587isinS119873

max1le119895le119873

dZ (119909120587minus1(119895) 119910119895) (132)




119892V(x 120596) + U(x) where

U (x) = sum119894 =119895

119880(d (119909119894 119909

119895)) 119880 N 997888rarr R (133)


V (x 120596) = 119881 (1199091 120596) + sdot sdot sdot + 119881 (119909

119873 120596) (134)






Appendix



V (1199091 119909

119873 120596) = sum

119910isinxn119910119881 (119910 120596) (A1)





119908(y)) 119881 (119908 120596)

+ sum

V =119908

119888V119881 (V 120596) (A2)


119888V = nV (x) minus nV (y) (A3)



119908 while

n119908(x) minus n

119908(y) = 119888

119908isin Z 0 hence 1003816100381610038161003816119888119908

1003816100381610038161003816 ge 1 (A4)

Therefore

P 1003816100381610038161003816119892V (x 120596) minus 119892V (y 120596)

1003816100381610038161003816 le 119904

= E [P 1003816100381610038161003816V (x 120596) minus V (y 120596)

1003816100381610038161003816 le10038161003816100381610038161198921003816100381610038161003816

minus1119904 | F

=119908]

= E [P 1003816100381610038161003816119888119908119881 (119908 120596) | z (120596)

1003816100381610038161003816 le10038161003816100381610038161198921003816100381610038161003816

minus1119904 | F

=119908]

(A5)




120596 119888119908119881 (119908 120596) isin [z (120596) minus

10038161003816100381610038161198921003816100381610038161003816

minus1119904 z (120596) +

10038161003816100381610038161198921003816100381610038161003816

minus1119904]

= 120596 119881 (119908 120596)

isin [z (120596) minus11990410038161003816100381610038161198921003816100381610038161003816

minus1

119888119908

z (120596) +11990410038161003816100381610038161198921003816100381610038161003816

minus1

119888119908

]

(A6)


P 1003816100381610038161003816V (x 120596) minus V (y 120596)

1003816100381610038161003816 le10038161003816100381610038161198921003816100381610038161003816

minus1119904

le sup119886isinR

(119865119881119908(119886 + 2119904) minus 119865

119881119908(119886))

(A7)


at 119908 given F=119908 Since 119865




1003816100381610038161003816 le 119904 (A8)


1198710(u) admits



0 To prove the




100381610038161003816100381610038161003816GB1198710 (u)

(x y 119864)100381610038161003816100381610038161003816le eminus119898(119904)d(xy)

(A9)





119899

119895=11198612119871(119909

119895) If




1198711+120575(x) can be


(⋃

119895isinJ

1198612119871(119909

119895))∐(⋃

119894isinJ119888

1198612119871(119909

119894)) (A10)


120588 (119861119871(119909

119894) 119861

119871(119909

119895)) gt 2119871 (A11)


119873119871 then for some 119895

∘ 120588(119909

119895∘ 119910

119895∘) gt 119862

119873119871


120588 (119909119895 119909

119895∘) le diamΠx le 119860

119873119871

120588 (119910119895 119910

119895∘) le diamΠy lt 119860

119873119871

(A12)


120588 (119909119894 119910

119895) ge 120588 (119909

119895∘ 119910

119895∘) minus 120588 (119909

119894 119909

119895∘) minus 120588 (119910

119895 119910

119895∘)

ge 119862119873119871 minus 2119860

119873119871 ge 3119871

(A13)


yielding

120588 (Π119861119871(x) Π119861

119871(y)) gt 3119871 minus 2119871 = 119871 (A14)



Acknowledgments


References

































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











1003816100381610038161003816GB (x y 119864)1003816100381610038161003816 le sum

119864119895isinΣ(HB)

10038161003816100381610038161003816120595

119895(x)1003816100381610038161003816100381610038161003816100381610038161003816120595

119895(y)10038161003816100381610038161003816

10038161003816100381610038161003816119864 minus 119864

119895

10038161003816100381610038161003816

(53)


120573


1003816100381610038161003816GB (x y 119864)1003816100381610038161003816 le eminus120574(119898119871)119871+119871

120573+ln|B|

le eminus120574(119898119871)119871+2119871120573

(54)



119898(1 + 119871minus120591

119896) 119871

119896minus 2119871

120573

119896= 119898119871

119896+ 119898119871

1minus120591

119896minus 2119871

120573

119896

ge 119898(1 +1

2119871minus120591

119896) 119871

119896

(55)




(1)




(119864119898)-NS(B) if a ball B


(119864119898)-NS




119871119896(k) sub B

119871119896(u) with






120588 (xB(119862119873minus1)119871119896

(w)) + 120588 (yB(119862119873minus1)119871119896(w))

ge 119877 minus (2119862119873minus 2) 119871

119896

(56)

Assume first that

1199031015840= 120588 (xB

(119862119873minus1)119871119896(w)) ge 119871

119896+ 1

11990310158401015840= 120588 (yB

(119862119873minus1)119871119896(w)) ge 119871

119896+ 1

(57)



119896are auto-


Furthermore

1199031015840+ 119903

10158401015840ge 119877 minus 2 (119862

119873minus 1) 119871

119896minus 2 ge 119877 minus 2119862

119873119871119896 (58)










(119877minus2119862119873119871119896minus2119871119896)(119871119896+1)

e119871120573

119896+1]

= 119898(1 +1

2119871minus120591

119896)119871119896

119871119896+ 1119877 (1 minus 3C

119873119877minus1119871119896) minus 119871

120573

119896+1

= 119898119877[(1 +1

2119871minus120591

119896) (1 minus 119871

minus1

119896) (1 minus 3119862

119873119871minus984858

119896)

minus119871120573

119896+1

119898119877] ge 119898(1 +

1

4119871minus120591

119896)119877 ge 120574 (119898 119871

119896+1) 119877

(59)




B119871119896(x) B


119871119896+1(u) is 119864-


119871119896(y) must


the hypothesis






119895 (k1015840 k10158401015840) 997891rarr |120595

119895(k1015840)120595

119895(k10158401015840)| where

120595119895 119895 isin [1 |B





119895 = 1 so the


119871119896+1(u) and

assume 120588(x y) = 119877 ge 1198711+984858119896




minus ln 10038161003816100381610038161003816120595119895 (x)120595119895 (y)10038161003816100381610038161003816


(119877minus2119862119873119871119896minus2119871119896)(119871119896+1)

]

(60)






120588 (x y) ge 1198711+984858119896minus1

0 lt 120591 lt 120588

1 + 984858 lt 120572

120573 lt 1 minus 120591

(61)


120572 =4

3

120588 =1

6

120591 =1

8

120573 =1

2

(62)




119901 gt 0 there is 1198920= 119892

0(119898 119901 119871


|119892| ge 1198920 any ball B

1198710(119906) and any 119864 isin R

P B1198710(u) 119894119904 (119864119898) -119878 le 119871minus119901

0

P B1198710(u) 119894119904 119898-NL119900119888 le 119871

minus119901

0

(63)




1015840isin (0 1] 1198601015840


119871(u1015840) B(119873)



P dist (Σ119871u1015840 Σ119871u10158401015840) le 119904 le (3119871)

2119873ℎ119871 (2119904) (64)


HB119871(u10158401015840) and

ℎ119871 (119904) = 119862

1015840101584011987111986010158401015840

119904119887and1198871015840

119887 and 1198871015840equiv min [119887 1198871015840] (65)

and 11986010158401015840= 119860

10158401015840(119860



u10158401015840) gt 4119873119871

P exist119864 isin R B119871(u1015840) B

119871(u10158401015840) 119886119903119890 119864-119875119877 le eminus119871

1205731015840

(66)





119873= 4119873 and





H(119873)

B119871(u) = H(1198991015840)

B119871(u1015840)otimes 1(119899

10158401015840)+ 1(119899

1015840)otimesH(119899

10158401015840)

B119871(u10158401015840)(67)

with 1198991015840+11989910158401015840 = 119873 u = (u1015840 u10158401015840) u1015840isinZn1015840

gt u10158401015840isinZn10158401015840

gt while for




(x) ΠJ119888B119871(x)) gt 2119871

(B) If B119871(x) B


then 120588(ΠB119871(x) ΠB

119871(y)) ge 2119871






B(119873)

119871119896(u) = B(119899

1015840)

119871119896(u1015840) times B(119899

10158401015840)

119871119896(u10158401015840) (68)


If the balls B(1198991015840)

119871119896minus1(u1015840) and B(119899

10158401015840)


B(119873)

119871119896(u) is also119898-Loc



HB119871119896 (u)= HB(119899

1015840)119871119896

(u1015840) otimes 1(11989910158401015840)+ 1(119899

1015840)otimesHB(119899

10158401015840)119871119896

(u10158401015840) (69)



119895= 120601

1198951015840 otimes120595

11989510158401015840 where 120601


1015840)119871119896



10158401015840)119871119896

(u10158401015840) Now


119895 (x)Ψ119895 (y)10038161003816100381610038161003816

=100381610038161003816100381610038161206011198951015840 (x1015840)120601

11989510158401015840 (x10158401015840)120595

1198951015840 (y1015840)120595

11989510158401015840 (y10158401015840)10038161003816100381610038161003816

(70)


1015840)

119871119896(u1015840) and B(119899

10158401015840)

119871119896(u10158401015840)


119875 (119899 119896) = 2119873minus119899119901 (1 + 120579)

119896 1 le 119899 le 119873 119896 ge 0 (71)



119871119896(u)

P B(119899)

119871119896(u) 119894119904 119898-NL119900119888 le 119871

minus119875(119899119896)

119896 (72)


119871119896(u) one has

P B(119873)

119871119896(u) 119894119904 119898-NL119900119888 le 2119871

minus2119875(119873119896)

119896le1

4119871minus119875(119873119896)

119896 (73)




1015840)

119871119896(u1015840) B(119899

10158401015840)

119871119896(u10158401015840) is 119898-NLoc


P B(119873)

119871119896(u) is 119898-NL119900119888

le P B(11989910158401015840)

119871119896(u10158401015840) is 119898-NL119900119888

+ P B(11989910158401015840)

119871119896(u10158401015840) is 119898-NL119900119888

le 119871minus119875(1198991015840119896)

119896+ 119871

minus119875(11989910158401015840119896)

119896le 2119871

minus2119875(119873119896)

119896lt1

4119871minus119875(119873119896)

119896

(74)





1198961015840 with 1198961015840 le 119896


119871119896(u)

P B(119873)

119871 119894(u) 119894119904 119898-NL119900119888 le 119871

minus119875(119873119894)

119894 (75)


119871119896+1(x) one has

P B(119873)

119871119896+1(u) 119894119904 119898-NL119900119888 le 119871

minus119875(119873119896+1)

119896+1 (76)

provided that

119901 gt2120572

2

2 minus 1205722119873119889

0 lt 3120579 le min2 minus 1205722

1205722minus2119873119889


(77)


119871119896(u) B(119873)

119871119896(k) one

has for some 119886 119888 gt 0


119871119896(u) B(119873)

119871119896(k) 119886119903119890 (119864119898) -119878

le 119871minus119875(119873119896)

119896equiv 119871

minus119901(1+120599)119896


1+119888119871119896

(78)


N119896+1= B(119873)

119871119896+1(u) is 119898-NLo119888

S(2)

119896


119871119896(u) B(119873)

119871119896(k)

sub B(119873)

119871119896+1(w)

R(2)


119871119896(u) B(119873)

119871119896(k)

sub B(119873)

119871119896+1(w)

(79)


N119896+1sub S

(2)

119896subR

(2)

119896cup (S

(2)

119896R

(2)

119896) (80)


119896 le eminus119871

1205731015840


119896R

(2)


119871119896+1(w) and


S(2)

119896(u k) = exist119864 B(119873)

119871119896(u) B(119873)

119871119896(k) are (119864119898) -S (81)


119896(u k) R(2)

119896 either B(119873)

119871119896(u) or B(119873)

119871119896(k)


119871119896(u) is 119864-CNR



pair of 119898-NL119900119888 balls B(119873)

119871119896minus1(y1015840) B(119873)


(12)1198711+120575


(1) Both B(119873)

119871119896minus1(y1015840) and B(119873)

119871119896minus1(y10158401015840) are FI


120588 (ΠB119871119896minus1(u) ΠB

119871119896minus1(k)) ge 2119871

119896 (82)


119896minus1= 119871

11205722

119896+1and 119871

0large enough

P B(119873)

119871119896minus1(y1015840) B(119873)

119871119896minus1(y10158401015840) are 119898-NL119900119888

le 2119871minus(2119901120572

2)(1+120579)

119896minus1

119896+1

(83)


119871119896(w)|2 le (321198732)1198712119873


P S(2)

119896R

(2)

119896 le32119873

2119871minus(2119901120572

2)(1+120579)

119896minus1+2119873

119896+1 (84)



119896+1 under con-


2119901

12057222119873minus119873119901 (1 + 120579)

119896minus1minus 2119873119889 gt 2

119873minus119873119901 (1 + 120579)

119896+1 (85)

or equivalently

(1 + 120579)2lt2119901

1205722minus

2119873119889

119901 (1 + 120579)119896minus1 (86)


3120599 le2 minus 120572

2

1205722minus2119873

119901 (87)


P S(2)

119896 le P R

(2)

119896 + P S

(2)

119896R

(2)

119896

le eminus1198711205731015840

119896 +1

4119871minus119875(119873119896+1)

119896+1lt1

2119871minus119875(119873119896+1)

119896+1

(88)


119871119896minus1(y1015840) orB(119873)


P B(119873)

119871119896minus1(y1015840) B(119873)

119871119896minus1(y10158401015840) is 119898-NL119900119888

le min (P B(119873)

119871119896minus1(y1015840) is 119898-NL119900119888

P B(119873)

119871119896minus1(y10158401015840) is 119898-NL119900119888) le 119871

minus2119901(1+120579)119896minus1

119896minus1

lt 119871minus119901(1+120579)

119896+1

119896+1

(89)



119871minus119875(119873119896)

119896equiv 119871

minus119901(1+120599)119896


1+119888119871119896 (90)




P B(119873)

119871119896(u) 119894119904 119898-NL119900119888 le

1

4119871minus119901(1+120579)

119896

119896(91)


minus119901(1+120579)119896


119895of operator HB(119873)119871119896 (u)



119896one

has10038161003816100381610038161003816Ψ119895(x) Ψ






1003816100381610038161003816Ψ (y)1003816100381610038161003816 le eminus119898y

(93)








119871119896(y) sub

B119871119896+2(z) Fix 120596 isin Ω1015840


2le 1 implies Ψ

infinle 1 there is


119899isin B

119871119896minus1(0) then






119871119896+2(z) with 120588(y z) isin [119871

119896+1 119871

119896+2)


1+120575

119896) is (120582119898)-NS Note that



119877(y) with

119877 = 120588(y z)minus21198711+120575119896minus1 gt (12)119871

1+2120575



minusln 1003816100381610038161003816Ψ (y)

1003816100381610038161003816

120588 (y z)ge 119898 (1 + 119871

minus120591

119896) (1 minus

2119871119896+ 1

120588 (y z))

ge 119898(1 +1

2119871minus120591

119896) gt 119898

(95)









119871(119909) B


P exist 119864 isin 119868 B119871(x) B

119871(y) 119886119903119890 (119864119898) -S le 119891 (119871) (96)


119871(119909) cup B

119871(119910) one has

E[ sup120601isinB1(119868)

10038161003816100381610038161003816⟨1x1003816100381610038161003816120601 (HΛ (120596))

1003816100381610038161003816 1y⟩10038161003816100381610038161003816]

le 119862119871119889119890minus119898119871+ 119891 (119871)

(97)




Λ sup B119871(119909) cup B

119871(119910)


10038161003816100381610038161003816⟨1x1003816100381610038161003816120601 (HΛ (120596))

1003816100381610038161003816 1y⟩10038161003816100381610038161003816] le 119862119890

minus119886 ln1+119888120588(xy) (98)










same 119888 119886 gt 0 as in (98)

E[ sup120601isinC1(119868)

10038161003816100381610038161003816⟨1x1003816100381610038161003816120601 (H (120596))

1003816100381610038161003816 1y⟩10038161003816100381610038161003816] le 119862eminus119886 ln

1+119888120588(xy) (99)




int120601 (120582) 119889120583xyB120596 (120582)

= ⟨1x1003816100381610038161003816 120601 (HB (120596))Π119868

(HB (120596))100381610038161003816100381610038161y⟩

(100)








119899rarr H with







119871119896

E [1003816100381610038161003816120583

xy120596


E [100381610038161003816100381610038161003816120583xyB119871119896

100381610038161003816100381610038161003816(E)] (101)




120601119905 120582 997891997888rarr 119891

119868(120582) ei119905120582 119905 isin R (102)








984858 = 2120575

120572 = 1 + 4120575

120591 =120575

2

(103)



120574 (119898 119871 119899) = 119898 (1 + 119871minus120591)119873minus119899+1

(104)



120588(x y) ge 1198621198731198711+120575 with 119862

119873= 4119873






119871(x) admits

a factorization

B(119873)

119871(x) = B(119899

1015840)

119871(x1015840) times B(119899

10158401015840)

119871(x10158401015840) (105)


119871(x) B


then 120588(ΠB119871(x) ΠB

119871(y)) ge 1198711+120575




119877119877 ge 119903



119880(2)

119877(119903) = 1

119903le119877119880

(2)(119903) (106)


gt



119877(x1015840) + U



11987110158401015840(x10158401015840) with

120588(B1198711015840(x1015840)B



H(119877)

B = H(119877)

B1198711015840(x1015840) otimes 1

(11989910158401015840)+ 1(119899

1015840)otimesH(119877)

B11987110158401015840

(x10158401015840) (107)


B(119873)


119873= 4119898119873


119871(x) =

B1198711015840(x1015840) times B

11987110158401015840(x10158401015840) with dist(x1015840 x10158401015840) gt (119862

1198732119873)119871 = 2119898119871

so that with 119877119896= (119862

1198732119873)119871

119896 one has

120598 (119877119896) lt eminus2119898119871119896 lt

1

2eminus119871120573

119896 (108)


119871119896(u) =

B(1198991015840)

119871119896(uJ)timesB

(11989910158401015840)



(a) dist(x1015840 x10158401015840) gt 119877119896= (119862

1198732119873)119871

119896

(b) B(1198991015840)

119871119896(u1015840) and B(119899

10158401015840)

119871119896(u10158401015840) are119898-Loc

(c) B(119873)

119871119896(u) is 119864-NR


119871119896(u) is (119864119898)-NS

Proof Set 119877119896= (119862

1198732119873)119871


above Denote

H(119877119896119873)

u119896 = H(119877119896)

B(119873)119871119896 (u)

G (119864) = (HB(119873)119871119896 (u)minus 119864)

minus1

(109)

and G(119877119896)(119864) = (H(119877119896119873)



H(119877119896 119873)

u119896 = HB(1198991015840)119871119896

(u1015840) otimes 1(11989910158401015840)+ 1(119899

1015840)otimesHB(119899

1015840)119871119896

(x10158401015840) (110)

with u = (u1015840 u10158401015840) isin Z119889119899

1015840

times Z11988911989910158401015840


119886119887= 120582

119886+120583

119887 where 120582

119886 = Σ(HB119871119896 (u



119887 = Σ(HB119871119896 (u


of H(119877119896 119873)


119886otimes 120595

119887 where


1015840)119871119896


of HB(11989910158401015840)119871119896


119886+ 120583

119887




|119864 minus (120582119886+ 120583

119887)| ge e119871

120573


119887)| ge e119871

120573

119896 and alsoas |(119864 minus 120583

119887) minus 120582

119886)| ge e119871

120573


G(119877119896) (u y 119864)

= sum

120582119886

sum

120583119887

120601119886(u1015840)120601

119886(y1015840)120595

119887(u10158401015840)120595

119887(y10158401015840)

(120582119886+ 120583

119887) minus 119864

= sum

120582119886

P1015840

119886(u1015840 y1015840)G(119877119896)

B119871119896 (u10158401015840)(u10158401015840 y10158401015840 119864 minus 120582

119886)

= sum

120583119887

P10158401015840

119887(u10158401015840 y10158401015840)G(119877119896)

B119871119896 (u1015840)(u1015840 y1015840 119864 minus 120583

119887)

(111)


B119871119896 (u1015840)(119864minus120583

119887) andG(119877119896)

B119871119896 (u10158401015840)(119864minus120582

119886) are

nonresonant1003817100381710038171003817100381710038171003817G(119877119896)

B119871119896 (u1015840)(119864 minus 120583

119887)

1003817100381710038171003817100381710038171003817le e119871

120573

119896

1003817100381710038171003817100381710038171003817G(119877119896)

B119871119896 (u10158401015840)(119864 minus 120582

119886)

1003817100381710038171003817100381710038171003817le e119871

120573

119896

(112)


y1015840) = 119871119896 in which case we

infer from (b)

10038161003816100381610038161003816G(119877119896) (u y 119864)10038161003816100381610038161003816 le

10038161003816100381610038161003816B119871119896(u10158401015840)10038161003816100381610038161003816eminus120574(119898119871119896119873minus1)119871119896+119871

120573

119896 (113)

or 120588(u10158401015840 y10158401015840) = 119871


10038161003816100381610038161003816G(119877119896) (u y 119864)10038161003816100381610038161003816 le

10038161003816100381610038161003816B119871119896(u1015840)10038161003816100381610038161003816eminus120574(119898119871119896119873minus1)119871119896+119871

120573

119896 (114)



119871119896+ 119871

120573

119896+ 119862 ln 119871

119896)

lt1

2eminus120574(119898119871119896119873)

(115)




10038171003817100381710038171003817G minus G(119877119896)

10038171003817100381710038171003817le10038171003817100381710038171003817U minus U(119877119896)

10038171003817100381710038171003817

10038171003817100381710038171003817G(119877119896)

10038171003817100381710038171003817G le eminus2119898119871119896e2119871

120573

119896

le1

2eminus120574(119898119871119896119873)

(117)


10038161003816100381610038161003816G (x y 119864) minus G(119877119896) (x y 119864)10038161003816100381610038161003816 le

1

2eminus120574(119898119871119896 119873)


119871119896+1(u)

(118)

Lemma 33 Let B119871119896+1(x)B



P B119871119896+1(x) B

119871119896+1(y) 119886119903119890 (119864119898) -119878

lt1

4119871minus119875(119873119896+1)

119896+1

(119)

Proof Let

R(2)= exist119864 B

119871119896+1(x) B

119871119896+1(y) 119886119903119890 119864-R

S(2)= B

119871119896+1(x) B119871119896+1

(y) 119886119903119890 (119864119898) -S (120)

ThenPS(2) le PR(2)

+PS(2)R(2)


119871119896+1(x) B

119871119896+1(y) must be 119864-NR


B(119873)

119871119896+1(x) = B(119899

1015840)

119871119896+1(x1015840) times B(119899

1015840)

119871119896+1(x10158401015840) (121)


119871119896+1(x10158401015840) are119898-Loc and

B119871119896+1(x) is (119864119898)-S then B




B(119873)

119871119896+1(y) = B(119899

1015840)

119871119896+1(y1015840) times B(119899

10158401015840)

119871119896+1(y10158401015840) (122)

If both B(1198991015840)119871119896+1(y1015840) and B(11989910158401015840)

119871119896+1(y10158401015840) are 119898-Loc and

B(119873)

119871119896+1(y) is (119864119898)-S then B(119873)

119871119896+1(y)must be 119864-R Let


1015840)

119871119896+1(x1015840) B(119899

10158401015840)

119871119896+1(x10158401015840)

B(1198991015840)

119871119896+1(y1015840) B(119899

1015840)

119871119896+1(y10158401015840) is 119898-NL119900119888

(123)


P L(4) le 4119871

minus119875(119873minus1119896+1)

119896+1= 4119871

minus2119875(119873119896+1)

119896+1 (124)



large enough

P S(2) le P R

(2) + P L

(4)

le eminus119871120573

119896 + 4119871minus2119875(119873119896+1)

119896+1lt1

4119871minus119875(119873119896+1)

119896+1

(125)






B119871119896(k) sub B

119871119896+1(u)with 120588(w k) ge 119862

1198731198711+120575

119896is (119864119898)-NS In the







119896also large)

1198711+984858

119896≫ 119871

1+120575


120588 (xB(119862119873minus1)119871119896

(w)) + 120588 (yB(119862119873minus1)119871119896(w))

ge 119877 minus (2119862119873minus 2) 119871

1+120575

119896

(126)

Assume first that

1199031015840= 120588 (xB

(119862119873minus1)1198711+120575119896(w)) ge 119871

119896+ 1

11990310158401015840= 120588 (yB

(119862119873minus1)1198711+120575119896(w)) ge 119871119896 + 1

(127)


1199031015840(x) and in B

11990310158401015840(y) are


1)1198711+120575

119896minus 2 ge 119877 minus 2119862

1198731198711+120575

119896










(119877minus21198621198731198711+120575119896 minus2119871119896)(119871119896+1)

sdot e119871120573

119896+1] = 119898(1 +1

2119871minus120591

119896)

119869119871119896

119871119896+ 1119877(1 minus

3119862119873

119877

sdot 119871119896) minus 119871

120573

119896+1= 119898119877[(1 +

1

2119871minus120591

119896)

119869


119896)


119873119871119896

1198711+984858

119896

) minus119871120573

119896+1

119898119877] = 119898119877[(1 +

1

2119871minus120591

119896)

119869


119896) (1 minus 3119862

119873119871minus984858

119896) minus

119871120573

119896+1

1198981198711+984858

119896

] ge 119898(1 +1

4


119896)

119869

119877 ge 120574 (119898 119871119896+1) 119877

(128)






1 119909

119873) 119909





1003816100381610038161003816 le 119862119889119871119889 (129)





by the functions

Φa =1

radic119873

sum

120587isinS119873

119873

⨂

119895=1

1119886120587minus1(119895) (130)

where a = 1198861 119886

119873 with 119886

1 119886

119873 = 119873



set is

ZNgt

= a = 1198861 119886

119873 119886

119895isinZ 119886

1 119886

119873 = 119873

(131)



1 119888

2 119888

119873 b = 119887

1 119888

2 119888

119873 with

1198861 119887

1 119888

2 119888

119873 = 119873 + 1

(ii) dZ(1198861 1198871) = 1


120588 (x y) = min120587isinS119873

max1le119895le119873

dZ (119909120587minus1(119895) 119910119895) (132)




119892V(x 120596) + U(x) where

U (x) = sum119894 =119895

119880(d (119909119894 119909

119895)) 119880 N 997888rarr R (133)


V (x 120596) = 119881 (1199091 120596) + sdot sdot sdot + 119881 (119909

119873 120596) (134)






Appendix



V (1199091 119909

119873 120596) = sum

119910isinxn119910119881 (119910 120596) (A1)





119908(y)) 119881 (119908 120596)

+ sum

V =119908

119888V119881 (V 120596) (A2)


119888V = nV (x) minus nV (y) (A3)



119908 while

n119908(x) minus n

119908(y) = 119888

119908isin Z 0 hence 1003816100381610038161003816119888119908

1003816100381610038161003816 ge 1 (A4)

Therefore

P 1003816100381610038161003816119892V (x 120596) minus 119892V (y 120596)

1003816100381610038161003816 le 119904

= E [P 1003816100381610038161003816V (x 120596) minus V (y 120596)

1003816100381610038161003816 le10038161003816100381610038161198921003816100381610038161003816

minus1119904 | F

=119908]

= E [P 1003816100381610038161003816119888119908119881 (119908 120596) | z (120596)

1003816100381610038161003816 le10038161003816100381610038161198921003816100381610038161003816

minus1119904 | F

=119908]

(A5)




120596 119888119908119881 (119908 120596) isin [z (120596) minus

10038161003816100381610038161198921003816100381610038161003816

minus1119904 z (120596) +

10038161003816100381610038161198921003816100381610038161003816

minus1119904]

= 120596 119881 (119908 120596)

isin [z (120596) minus11990410038161003816100381610038161198921003816100381610038161003816

minus1

119888119908

z (120596) +11990410038161003816100381610038161198921003816100381610038161003816

minus1

119888119908

]

(A6)


P 1003816100381610038161003816V (x 120596) minus V (y 120596)

1003816100381610038161003816 le10038161003816100381610038161198921003816100381610038161003816

minus1119904

le sup119886isinR

(119865119881119908(119886 + 2119904) minus 119865

119881119908(119886))

(A7)


at 119908 given F=119908 Since 119865




1003816100381610038161003816 le 119904 (A8)


1198710(u) admits



0 To prove the




100381610038161003816100381610038161003816GB1198710 (u)

(x y 119864)100381610038161003816100381610038161003816le eminus119898(119904)d(xy)

(A9)





119899

119895=11198612119871(119909

119895) If




1198711+120575(x) can be


(⋃

119895isinJ

1198612119871(119909

119895))∐(⋃

119894isinJ119888

1198612119871(119909

119894)) (A10)


120588 (119861119871(119909

119894) 119861

119871(119909

119895)) gt 2119871 (A11)


119873119871 then for some 119895

∘ 120588(119909

119895∘ 119910

119895∘) gt 119862

119873119871


120588 (119909119895 119909

119895∘) le diamΠx le 119860

119873119871

120588 (119910119895 119910

119895∘) le diamΠy lt 119860

119873119871

(A12)


120588 (119909119894 119910

119895) ge 120588 (119909

119895∘ 119910

119895∘) minus 120588 (119909

119894 119909

119895∘) minus 120588 (119910

119895 119910

119895∘)

ge 119862119873119871 minus 2119860

119873119871 ge 3119871

(A13)


yielding

120588 (Π119861119871(x) Π119861

119871(y)) gt 3119871 minus 2119871 = 119871 (A14)



Acknowledgments


References

































Volume 2014




Journal of











Journal of


Function Spaces






Algebra













120588 (x y) ge 1198711+984858119896minus1

0 lt 120591 lt 120588

1 + 984858 lt 120572

120573 lt 1 minus 120591

(61)


120572 =4

3

120588 =1

6

120591 =1

8

120573 =1

2

(62)




119901 gt 0 there is 1198920= 119892

0(119898 119901 119871


|119892| ge 1198920 any ball B

1198710(119906) and any 119864 isin R

P B1198710(u) 119894119904 (119864119898) -119878 le 119871minus119901

0

P B1198710(u) 119894119904 119898-NL119900119888 le 119871

minus119901

0

(63)




1015840isin (0 1] 1198601015840


119871(u1015840) B(119873)



P dist (Σ119871u1015840 Σ119871u10158401015840) le 119904 le (3119871)

2119873ℎ119871 (2119904) (64)


HB119871(u10158401015840) and

ℎ119871 (119904) = 119862

1015840101584011987111986010158401015840

119904119887and1198871015840

119887 and 1198871015840equiv min [119887 1198871015840] (65)

and 11986010158401015840= 119860

10158401015840(119860



u10158401015840) gt 4119873119871

P exist119864 isin R B119871(u1015840) B

119871(u10158401015840) 119886119903119890 119864-119875119877 le eminus119871

1205731015840

(66)





119873= 4119873 and





H(119873)

B119871(u) = H(1198991015840)

B119871(u1015840)otimes 1(119899

10158401015840)+ 1(119899

1015840)otimesH(119899

10158401015840)

B119871(u10158401015840)(67)

with 1198991015840+11989910158401015840 = 119873 u = (u1015840 u10158401015840) u1015840isinZn1015840

gt u10158401015840isinZn10158401015840

gt while for




(x) ΠJ119888B119871(x)) gt 2119871

(B) If B119871(x) B


then 120588(ΠB119871(x) ΠB

119871(y)) ge 2119871






B(119873)

119871119896(u) = B(119899

1015840)

119871119896(u1015840) times B(119899

10158401015840)

119871119896(u10158401015840) (68)


If the balls B(1198991015840)

119871119896minus1(u1015840) and B(119899

10158401015840)


B(119873)

119871119896(u) is also119898-Loc



HB119871119896 (u)= HB(119899

1015840)119871119896

(u1015840) otimes 1(11989910158401015840)+ 1(119899

1015840)otimesHB(119899

10158401015840)119871119896

(u10158401015840) (69)



119895= 120601

1198951015840 otimes120595

11989510158401015840 where 120601


1015840)119871119896



10158401015840)119871119896

(u10158401015840) Now


119895 (x)Ψ119895 (y)10038161003816100381610038161003816

=100381610038161003816100381610038161206011198951015840 (x1015840)120601

11989510158401015840 (x10158401015840)120595

1198951015840 (y1015840)120595

11989510158401015840 (y10158401015840)10038161003816100381610038161003816

(70)


1015840)

119871119896(u1015840) and B(119899

10158401015840)

119871119896(u10158401015840)


119875 (119899 119896) = 2119873minus119899119901 (1 + 120579)

119896 1 le 119899 le 119873 119896 ge 0 (71)



119871119896(u)

P B(119899)

119871119896(u) 119894119904 119898-NL119900119888 le 119871

minus119875(119899119896)

119896 (72)


119871119896(u) one has

P B(119873)

119871119896(u) 119894119904 119898-NL119900119888 le 2119871

minus2119875(119873119896)

119896le1

4119871minus119875(119873119896)

119896 (73)




1015840)

119871119896(u1015840) B(119899

10158401015840)

119871119896(u10158401015840) is 119898-NLoc


P B(119873)

119871119896(u) is 119898-NL119900119888

le P B(11989910158401015840)

119871119896(u10158401015840) is 119898-NL119900119888

+ P B(11989910158401015840)

119871119896(u10158401015840) is 119898-NL119900119888

le 119871minus119875(1198991015840119896)

119896+ 119871

minus119875(11989910158401015840119896)

119896le 2119871

minus2119875(119873119896)

119896lt1

4119871minus119875(119873119896)

119896

(74)





1198961015840 with 1198961015840 le 119896


119871119896(u)

P B(119873)

119871 119894(u) 119894119904 119898-NL119900119888 le 119871

minus119875(119873119894)

119894 (75)


119871119896+1(x) one has

P B(119873)

119871119896+1(u) 119894119904 119898-NL119900119888 le 119871

minus119875(119873119896+1)

119896+1 (76)

provided that

119901 gt2120572

2

2 minus 1205722119873119889

0 lt 3120579 le min2 minus 1205722

1205722minus2119873119889


(77)


119871119896(u) B(119873)

119871119896(k) one

has for some 119886 119888 gt 0


119871119896(u) B(119873)

119871119896(k) 119886119903119890 (119864119898) -119878

le 119871minus119875(119873119896)

119896equiv 119871

minus119901(1+120599)119896


1+119888119871119896

(78)


N119896+1= B(119873)

119871119896+1(u) is 119898-NLo119888

S(2)

119896


119871119896(u) B(119873)

119871119896(k)

sub B(119873)

119871119896+1(w)

R(2)


119871119896(u) B(119873)

119871119896(k)

sub B(119873)

119871119896+1(w)

(79)


N119896+1sub S

(2)

119896subR

(2)

119896cup (S

(2)

119896R

(2)

119896) (80)


119896 le eminus119871

1205731015840


119896R

(2)


119871119896+1(w) and


S(2)

119896(u k) = exist119864 B(119873)

119871119896(u) B(119873)

119871119896(k) are (119864119898) -S (81)


119896(u k) R(2)

119896 either B(119873)

119871119896(u) or B(119873)

119871119896(k)


119871119896(u) is 119864-CNR



pair of 119898-NL119900119888 balls B(119873)

119871119896minus1(y1015840) B(119873)


(12)1198711+120575


(1) Both B(119873)

119871119896minus1(y1015840) and B(119873)

119871119896minus1(y10158401015840) are FI


120588 (ΠB119871119896minus1(u) ΠB

119871119896minus1(k)) ge 2119871

119896 (82)


119896minus1= 119871

11205722

119896+1and 119871

0large enough

P B(119873)

119871119896minus1(y1015840) B(119873)

119871119896minus1(y10158401015840) are 119898-NL119900119888

le 2119871minus(2119901120572

2)(1+120579)

119896minus1

119896+1

(83)


119871119896(w)|2 le (321198732)1198712119873


P S(2)

119896R

(2)

119896 le32119873

2119871minus(2119901120572

2)(1+120579)

119896minus1+2119873

119896+1 (84)



119896+1 under con-


2119901

12057222119873minus119873119901 (1 + 120579)

119896minus1minus 2119873119889 gt 2

119873minus119873119901 (1 + 120579)

119896+1 (85)

or equivalently

(1 + 120579)2lt2119901

1205722minus

2119873119889

119901 (1 + 120579)119896minus1 (86)


3120599 le2 minus 120572

2

1205722minus2119873

119901 (87)


P S(2)

119896 le P R

(2)

119896 + P S

(2)

119896R

(2)

119896

le eminus1198711205731015840

119896 +1

4119871minus119875(119873119896+1)

119896+1lt1

2119871minus119875(119873119896+1)

119896+1

(88)


119871119896minus1(y1015840) orB(119873)


P B(119873)

119871119896minus1(y1015840) B(119873)

119871119896minus1(y10158401015840) is 119898-NL119900119888

le min (P B(119873)

119871119896minus1(y1015840) is 119898-NL119900119888

P B(119873)

119871119896minus1(y10158401015840) is 119898-NL119900119888) le 119871

minus2119901(1+120579)119896minus1

119896minus1

lt 119871minus119901(1+120579)

119896+1

119896+1

(89)



119871minus119875(119873119896)

119896equiv 119871

minus119901(1+120599)119896


1+119888119871119896 (90)




P B(119873)

119871119896(u) 119894119904 119898-NL119900119888 le

1

4119871minus119901(1+120579)

119896

119896(91)


minus119901(1+120579)119896


119895of operator HB(119873)119871119896 (u)



119896one

has10038161003816100381610038161003816Ψ119895(x) Ψ






1003816100381610038161003816Ψ (y)1003816100381610038161003816 le eminus119898y

(93)








119871119896(y) sub

B119871119896+2(z) Fix 120596 isin Ω1015840


2le 1 implies Ψ

infinle 1 there is


119899isin B

119871119896minus1(0) then






119871119896+2(z) with 120588(y z) isin [119871

119896+1 119871

119896+2)


1+120575

119896) is (120582119898)-NS Note that



119877(y) with

119877 = 120588(y z)minus21198711+120575119896minus1 gt (12)119871

1+2120575



minusln 1003816100381610038161003816Ψ (y)

1003816100381610038161003816

120588 (y z)ge 119898 (1 + 119871

minus120591

119896) (1 minus

2119871119896+ 1

120588 (y z))

ge 119898(1 +1

2119871minus120591

119896) gt 119898

(95)









119871(119909) B


P exist 119864 isin 119868 B119871(x) B

119871(y) 119886119903119890 (119864119898) -S le 119891 (119871) (96)


119871(119909) cup B

119871(119910) one has

E[ sup120601isinB1(119868)

10038161003816100381610038161003816⟨1x1003816100381610038161003816120601 (HΛ (120596))

1003816100381610038161003816 1y⟩10038161003816100381610038161003816]

le 119862119871119889119890minus119898119871+ 119891 (119871)

(97)




Λ sup B119871(119909) cup B

119871(119910)


10038161003816100381610038161003816⟨1x1003816100381610038161003816120601 (HΛ (120596))

1003816100381610038161003816 1y⟩10038161003816100381610038161003816] le 119862119890

minus119886 ln1+119888120588(xy) (98)










same 119888 119886 gt 0 as in (98)

E[ sup120601isinC1(119868)

10038161003816100381610038161003816⟨1x1003816100381610038161003816120601 (H (120596))

1003816100381610038161003816 1y⟩10038161003816100381610038161003816] le 119862eminus119886 ln

1+119888120588(xy) (99)




int120601 (120582) 119889120583xyB120596 (120582)

= ⟨1x1003816100381610038161003816 120601 (HB (120596))Π119868

(HB (120596))100381610038161003816100381610038161y⟩

(100)








119899rarr H with







119871119896

E [1003816100381610038161003816120583

xy120596


E [100381610038161003816100381610038161003816120583xyB119871119896

100381610038161003816100381610038161003816(E)] (101)




120601119905 120582 997891997888rarr 119891

119868(120582) ei119905120582 119905 isin R (102)








984858 = 2120575

120572 = 1 + 4120575

120591 =120575

2

(103)



120574 (119898 119871 119899) = 119898 (1 + 119871minus120591)119873minus119899+1

(104)



120588(x y) ge 1198621198731198711+120575 with 119862

119873= 4119873






119871(x) admits

a factorization

B(119873)

119871(x) = B(119899

1015840)

119871(x1015840) times B(119899

10158401015840)

119871(x10158401015840) (105)


119871(x) B


then 120588(ΠB119871(x) ΠB

119871(y)) ge 1198711+120575




119877119877 ge 119903



119880(2)

119877(119903) = 1

119903le119877119880

(2)(119903) (106)


gt



119877(x1015840) + U



11987110158401015840(x10158401015840) with

120588(B1198711015840(x1015840)B



H(119877)

B = H(119877)

B1198711015840(x1015840) otimes 1

(11989910158401015840)+ 1(119899

1015840)otimesH(119877)

B11987110158401015840

(x10158401015840) (107)


B(119873)


119873= 4119898119873


119871(x) =

B1198711015840(x1015840) times B

11987110158401015840(x10158401015840) with dist(x1015840 x10158401015840) gt (119862

1198732119873)119871 = 2119898119871

so that with 119877119896= (119862

1198732119873)119871

119896 one has

120598 (119877119896) lt eminus2119898119871119896 lt

1

2eminus119871120573

119896 (108)


119871119896(u) =

B(1198991015840)

119871119896(uJ)timesB

(11989910158401015840)



(a) dist(x1015840 x10158401015840) gt 119877119896= (119862

1198732119873)119871

119896

(b) B(1198991015840)

119871119896(u1015840) and B(119899

10158401015840)

119871119896(u10158401015840) are119898-Loc

(c) B(119873)

119871119896(u) is 119864-NR


119871119896(u) is (119864119898)-NS

Proof Set 119877119896= (119862

1198732119873)119871


above Denote

H(119877119896119873)

u119896 = H(119877119896)

B(119873)119871119896 (u)

G (119864) = (HB(119873)119871119896 (u)minus 119864)

minus1

(109)

and G(119877119896)(119864) = (H(119877119896119873)



H(119877119896 119873)

u119896 = HB(1198991015840)119871119896

(u1015840) otimes 1(11989910158401015840)+ 1(119899

1015840)otimesHB(119899

1015840)119871119896

(x10158401015840) (110)

with u = (u1015840 u10158401015840) isin Z119889119899

1015840

times Z11988911989910158401015840


119886119887= 120582

119886+120583

119887 where 120582

119886 = Σ(HB119871119896 (u



119887 = Σ(HB119871119896 (u


of H(119877119896 119873)


119886otimes 120595

119887 where


1015840)119871119896


of HB(11989910158401015840)119871119896


119886+ 120583

119887




|119864 minus (120582119886+ 120583

119887)| ge e119871

120573


119887)| ge e119871

120573

119896 and alsoas |(119864 minus 120583

119887) minus 120582

119886)| ge e119871

120573


G(119877119896) (u y 119864)

= sum

120582119886

sum

120583119887

120601119886(u1015840)120601

119886(y1015840)120595

119887(u10158401015840)120595

119887(y10158401015840)

(120582119886+ 120583

119887) minus 119864

= sum

120582119886

P1015840

119886(u1015840 y1015840)G(119877119896)

B119871119896 (u10158401015840)(u10158401015840 y10158401015840 119864 minus 120582

119886)

= sum

120583119887

P10158401015840

119887(u10158401015840 y10158401015840)G(119877119896)

B119871119896 (u1015840)(u1015840 y1015840 119864 minus 120583

119887)

(111)


B119871119896 (u1015840)(119864minus120583

119887) andG(119877119896)

B119871119896 (u10158401015840)(119864minus120582

119886) are

nonresonant1003817100381710038171003817100381710038171003817G(119877119896)

B119871119896 (u1015840)(119864 minus 120583

119887)

1003817100381710038171003817100381710038171003817le e119871

120573

119896

1003817100381710038171003817100381710038171003817G(119877119896)

B119871119896 (u10158401015840)(119864 minus 120582

119886)

1003817100381710038171003817100381710038171003817le e119871

120573

119896

(112)


y1015840) = 119871119896 in which case we

infer from (b)

10038161003816100381610038161003816G(119877119896) (u y 119864)10038161003816100381610038161003816 le

10038161003816100381610038161003816B119871119896(u10158401015840)10038161003816100381610038161003816eminus120574(119898119871119896119873minus1)119871119896+119871

120573

119896 (113)

or 120588(u10158401015840 y10158401015840) = 119871


10038161003816100381610038161003816G(119877119896) (u y 119864)10038161003816100381610038161003816 le

10038161003816100381610038161003816B119871119896(u1015840)10038161003816100381610038161003816eminus120574(119898119871119896119873minus1)119871119896+119871

120573

119896 (114)



119871119896+ 119871

120573

119896+ 119862 ln 119871

119896)

lt1

2eminus120574(119898119871119896119873)

(115)




10038171003817100381710038171003817G minus G(119877119896)

10038171003817100381710038171003817le10038171003817100381710038171003817U minus U(119877119896)

10038171003817100381710038171003817

10038171003817100381710038171003817G(119877119896)

10038171003817100381710038171003817G le eminus2119898119871119896e2119871

120573

119896

le1

2eminus120574(119898119871119896119873)

(117)


10038161003816100381610038161003816G (x y 119864) minus G(119877119896) (x y 119864)10038161003816100381610038161003816 le

1

2eminus120574(119898119871119896 119873)


119871119896+1(u)

(118)

Lemma 33 Let B119871119896+1(x)B



P B119871119896+1(x) B

119871119896+1(y) 119886119903119890 (119864119898) -119878

lt1

4119871minus119875(119873119896+1)

119896+1

(119)

Proof Let

R(2)= exist119864 B

119871119896+1(x) B

119871119896+1(y) 119886119903119890 119864-R

S(2)= B

119871119896+1(x) B119871119896+1

(y) 119886119903119890 (119864119898) -S (120)

ThenPS(2) le PR(2)

+PS(2)R(2)


119871119896+1(x) B

119871119896+1(y) must be 119864-NR


B(119873)

119871119896+1(x) = B(119899

1015840)

119871119896+1(x1015840) times B(119899

1015840)

119871119896+1(x10158401015840) (121)


119871119896+1(x10158401015840) are119898-Loc and

B119871119896+1(x) is (119864119898)-S then B




B(119873)

119871119896+1(y) = B(119899

1015840)

119871119896+1(y1015840) times B(119899

10158401015840)

119871119896+1(y10158401015840) (122)

If both B(1198991015840)119871119896+1(y1015840) and B(11989910158401015840)

119871119896+1(y10158401015840) are 119898-Loc and

B(119873)

119871119896+1(y) is (119864119898)-S then B(119873)

119871119896+1(y)must be 119864-R Let


1015840)

119871119896+1(x1015840) B(119899

10158401015840)

119871119896+1(x10158401015840)

B(1198991015840)

119871119896+1(y1015840) B(119899

1015840)

119871119896+1(y10158401015840) is 119898-NL119900119888

(123)


P L(4) le 4119871

minus119875(119873minus1119896+1)

119896+1= 4119871

minus2119875(119873119896+1)

119896+1 (124)



large enough

P S(2) le P R

(2) + P L

(4)

le eminus119871120573

119896 + 4119871minus2119875(119873119896+1)

119896+1lt1

4119871minus119875(119873119896+1)

119896+1

(125)






B119871119896(k) sub B

119871119896+1(u)with 120588(w k) ge 119862

1198731198711+120575

119896is (119864119898)-NS In the







119896also large)

1198711+984858

119896≫ 119871

1+120575


120588 (xB(119862119873minus1)119871119896

(w)) + 120588 (yB(119862119873minus1)119871119896(w))

ge 119877 minus (2119862119873minus 2) 119871

1+120575

119896

(126)

Assume first that

1199031015840= 120588 (xB

(119862119873minus1)1198711+120575119896(w)) ge 119871

119896+ 1

11990310158401015840= 120588 (yB

(119862119873minus1)1198711+120575119896(w)) ge 119871119896 + 1

(127)


1199031015840(x) and in B

11990310158401015840(y) are


1)1198711+120575

119896minus 2 ge 119877 minus 2119862

1198731198711+120575

119896










(119877minus21198621198731198711+120575119896 minus2119871119896)(119871119896+1)

sdot e119871120573

119896+1] = 119898(1 +1

2119871minus120591

119896)

119869119871119896

119871119896+ 1119877(1 minus

3119862119873

119877

sdot 119871119896) minus 119871

120573

119896+1= 119898119877[(1 +

1

2119871minus120591

119896)

119869


119896)


119873119871119896

1198711+984858

119896

) minus119871120573

119896+1

119898119877] = 119898119877[(1 +

1

2119871minus120591

119896)

119869


119896) (1 minus 3119862

119873119871minus984858

119896) minus

119871120573

119896+1

1198981198711+984858

119896

] ge 119898(1 +1

4


119896)

119869

119877 ge 120574 (119898 119871119896+1) 119877

(128)






1 119909

119873) 119909





1003816100381610038161003816 le 119862119889119871119889 (129)





by the functions

Φa =1

radic119873

sum

120587isinS119873

119873

⨂

119895=1

1119886120587minus1(119895) (130)

where a = 1198861 119886

119873 with 119886

1 119886

119873 = 119873



set is

ZNgt

= a = 1198861 119886

119873 119886

119895isinZ 119886

1 119886

119873 = 119873

(131)



1 119888

2 119888

119873 b = 119887

1 119888

2 119888

119873 with

1198861 119887

1 119888

2 119888

119873 = 119873 + 1

(ii) dZ(1198861 1198871) = 1


120588 (x y) = min120587isinS119873

max1le119895le119873

dZ (119909120587minus1(119895) 119910119895) (132)




119892V(x 120596) + U(x) where

U (x) = sum119894 =119895

119880(d (119909119894 119909

119895)) 119880 N 997888rarr R (133)


V (x 120596) = 119881 (1199091 120596) + sdot sdot sdot + 119881 (119909

119873 120596) (134)






Appendix



V (1199091 119909

119873 120596) = sum

119910isinxn119910119881 (119910 120596) (A1)





119908(y)) 119881 (119908 120596)

+ sum

V =119908

119888V119881 (V 120596) (A2)


119888V = nV (x) minus nV (y) (A3)



119908 while

n119908(x) minus n

119908(y) = 119888

119908isin Z 0 hence 1003816100381610038161003816119888119908

1003816100381610038161003816 ge 1 (A4)

Therefore

P 1003816100381610038161003816119892V (x 120596) minus 119892V (y 120596)

1003816100381610038161003816 le 119904

= E [P 1003816100381610038161003816V (x 120596) minus V (y 120596)

1003816100381610038161003816 le10038161003816100381610038161198921003816100381610038161003816

minus1119904 | F

=119908]

= E [P 1003816100381610038161003816119888119908119881 (119908 120596) | z (120596)

1003816100381610038161003816 le10038161003816100381610038161198921003816100381610038161003816

minus1119904 | F

=119908]

(A5)




120596 119888119908119881 (119908 120596) isin [z (120596) minus

10038161003816100381610038161198921003816100381610038161003816

minus1119904 z (120596) +

10038161003816100381610038161198921003816100381610038161003816

minus1119904]

= 120596 119881 (119908 120596)

isin [z (120596) minus11990410038161003816100381610038161198921003816100381610038161003816

minus1

119888119908

z (120596) +11990410038161003816100381610038161198921003816100381610038161003816

minus1

119888119908

]

(A6)


P 1003816100381610038161003816V (x 120596) minus V (y 120596)

1003816100381610038161003816 le10038161003816100381610038161198921003816100381610038161003816

minus1119904

le sup119886isinR

(119865119881119908(119886 + 2119904) minus 119865

119881119908(119886))

(A7)


at 119908 given F=119908 Since 119865




1003816100381610038161003816 le 119904 (A8)


1198710(u) admits



0 To prove the




100381610038161003816100381610038161003816GB1198710 (u)

(x y 119864)100381610038161003816100381610038161003816le eminus119898(119904)d(xy)

(A9)





119899

119895=11198612119871(119909

119895) If




1198711+120575(x) can be


(⋃

119895isinJ

1198612119871(119909

119895))∐(⋃

119894isinJ119888

1198612119871(119909

119894)) (A10)


120588 (119861119871(119909

119894) 119861

119871(119909

119895)) gt 2119871 (A11)


119873119871 then for some 119895

∘ 120588(119909

119895∘ 119910

119895∘) gt 119862

119873119871


120588 (119909119895 119909

119895∘) le diamΠx le 119860

119873119871

120588 (119910119895 119910

119895∘) le diamΠy lt 119860

119873119871

(A12)


120588 (119909119894 119910

119895) ge 120588 (119909

119895∘ 119910

119895∘) minus 120588 (119909

119894 119909

119895∘) minus 120588 (119910

119895 119910

119895∘)

ge 119862119873119871 minus 2119860

119873119871 ge 3119871

(A13)


yielding

120588 (Π119861119871(x) Π119861

119871(y)) gt 3119871 minus 2119871 = 119871 (A14)



Acknowledgments


References

































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










If the balls B(1198991015840)

119871119896minus1(u1015840) and B(119899

10158401015840)


B(119873)

119871119896(u) is also119898-Loc



HB119871119896 (u)= HB(119899

1015840)119871119896

(u1015840) otimes 1(11989910158401015840)+ 1(119899

1015840)otimesHB(119899

10158401015840)119871119896

(u10158401015840) (69)



119895= 120601

1198951015840 otimes120595

11989510158401015840 where 120601


1015840)119871119896



10158401015840)119871119896

(u10158401015840) Now


119895 (x)Ψ119895 (y)10038161003816100381610038161003816

=100381610038161003816100381610038161206011198951015840 (x1015840)120601

11989510158401015840 (x10158401015840)120595

1198951015840 (y1015840)120595

11989510158401015840 (y10158401015840)10038161003816100381610038161003816

(70)


1015840)

119871119896(u1015840) and B(119899

10158401015840)

119871119896(u10158401015840)


119875 (119899 119896) = 2119873minus119899119901 (1 + 120579)

119896 1 le 119899 le 119873 119896 ge 0 (71)



119871119896(u)

P B(119899)

119871119896(u) 119894119904 119898-NL119900119888 le 119871

minus119875(119899119896)

119896 (72)


119871119896(u) one has

P B(119873)

119871119896(u) 119894119904 119898-NL119900119888 le 2119871

minus2119875(119873119896)

119896le1

4119871minus119875(119873119896)

119896 (73)




1015840)

119871119896(u1015840) B(119899

10158401015840)

119871119896(u10158401015840) is 119898-NLoc


P B(119873)

119871119896(u) is 119898-NL119900119888

le P B(11989910158401015840)

119871119896(u10158401015840) is 119898-NL119900119888

+ P B(11989910158401015840)

119871119896(u10158401015840) is 119898-NL119900119888

le 119871minus119875(1198991015840119896)

119896+ 119871

minus119875(11989910158401015840119896)

119896le 2119871

minus2119875(119873119896)

119896lt1

4119871minus119875(119873119896)

119896

(74)





1198961015840 with 1198961015840 le 119896


119871119896(u)

P B(119873)

119871 119894(u) 119894119904 119898-NL119900119888 le 119871

minus119875(119873119894)

119894 (75)


119871119896+1(x) one has

P B(119873)

119871119896+1(u) 119894119904 119898-NL119900119888 le 119871

minus119875(119873119896+1)

119896+1 (76)

provided that

119901 gt2120572

2

2 minus 1205722119873119889

0 lt 3120579 le min2 minus 1205722

1205722minus2119873119889


(77)


119871119896(u) B(119873)

119871119896(k) one

has for some 119886 119888 gt 0


119871119896(u) B(119873)

119871119896(k) 119886119903119890 (119864119898) -119878

le 119871minus119875(119873119896)

119896equiv 119871

minus119901(1+120599)119896


1+119888119871119896

(78)


N119896+1= B(119873)

119871119896+1(u) is 119898-NLo119888

S(2)

119896


119871119896(u) B(119873)

119871119896(k)

sub B(119873)

119871119896+1(w)

R(2)


119871119896(u) B(119873)

119871119896(k)

sub B(119873)

119871119896+1(w)

(79)


N119896+1sub S

(2)

119896subR

(2)

119896cup (S

(2)

119896R

(2)

119896) (80)


119896 le eminus119871

1205731015840


119896R

(2)


119871119896+1(w) and


S(2)

119896(u k) = exist119864 B(119873)

119871119896(u) B(119873)

119871119896(k) are (119864119898) -S (81)


119896(u k) R(2)

119896 either B(119873)

119871119896(u) or B(119873)

119871119896(k)


119871119896(u) is 119864-CNR



pair of 119898-NL119900119888 balls B(119873)

119871119896minus1(y1015840) B(119873)


(12)1198711+120575


(1) Both B(119873)

119871119896minus1(y1015840) and B(119873)

119871119896minus1(y10158401015840) are FI


120588 (ΠB119871119896minus1(u) ΠB

119871119896minus1(k)) ge 2119871

119896 (82)


119896minus1= 119871

11205722

119896+1and 119871

0large enough

P B(119873)

119871119896minus1(y1015840) B(119873)

119871119896minus1(y10158401015840) are 119898-NL119900119888

le 2119871minus(2119901120572

2)(1+120579)

119896minus1

119896+1

(83)


119871119896(w)|2 le (321198732)1198712119873


P S(2)

119896R

(2)

119896 le32119873

2119871minus(2119901120572

2)(1+120579)

119896minus1+2119873

119896+1 (84)



119896+1 under con-


2119901

12057222119873minus119873119901 (1 + 120579)

119896minus1minus 2119873119889 gt 2

119873minus119873119901 (1 + 120579)

119896+1 (85)

or equivalently

(1 + 120579)2lt2119901

1205722minus

2119873119889

119901 (1 + 120579)119896minus1 (86)


3120599 le2 minus 120572

2

1205722minus2119873

119901 (87)


P S(2)

119896 le P R

(2)

119896 + P S

(2)

119896R

(2)

119896

le eminus1198711205731015840

119896 +1

4119871minus119875(119873119896+1)

119896+1lt1

2119871minus119875(119873119896+1)

119896+1

(88)


119871119896minus1(y1015840) orB(119873)


P B(119873)

119871119896minus1(y1015840) B(119873)

119871119896minus1(y10158401015840) is 119898-NL119900119888

le min (P B(119873)

119871119896minus1(y1015840) is 119898-NL119900119888

P B(119873)

119871119896minus1(y10158401015840) is 119898-NL119900119888) le 119871

minus2119901(1+120579)119896minus1

119896minus1

lt 119871minus119901(1+120579)

119896+1

119896+1

(89)



119871minus119875(119873119896)

119896equiv 119871

minus119901(1+120599)119896


1+119888119871119896 (90)




P B(119873)

119871119896(u) 119894119904 119898-NL119900119888 le

1

4119871minus119901(1+120579)

119896

119896(91)


minus119901(1+120579)119896


119895of operator HB(119873)119871119896 (u)



119896one

has10038161003816100381610038161003816Ψ119895(x) Ψ






1003816100381610038161003816Ψ (y)1003816100381610038161003816 le eminus119898y

(93)








119871119896(y) sub

B119871119896+2(z) Fix 120596 isin Ω1015840


2le 1 implies Ψ

infinle 1 there is


119899isin B

119871119896minus1(0) then






119871119896+2(z) with 120588(y z) isin [119871

119896+1 119871

119896+2)


1+120575

119896) is (120582119898)-NS Note that



119877(y) with

119877 = 120588(y z)minus21198711+120575119896minus1 gt (12)119871

1+2120575



minusln 1003816100381610038161003816Ψ (y)

1003816100381610038161003816

120588 (y z)ge 119898 (1 + 119871

minus120591

119896) (1 minus

2119871119896+ 1

120588 (y z))

ge 119898(1 +1

2119871minus120591

119896) gt 119898

(95)









119871(119909) B


P exist 119864 isin 119868 B119871(x) B

119871(y) 119886119903119890 (119864119898) -S le 119891 (119871) (96)


119871(119909) cup B

119871(119910) one has

E[ sup120601isinB1(119868)

10038161003816100381610038161003816⟨1x1003816100381610038161003816120601 (HΛ (120596))

1003816100381610038161003816 1y⟩10038161003816100381610038161003816]

le 119862119871119889119890minus119898119871+ 119891 (119871)

(97)




Λ sup B119871(119909) cup B

119871(119910)


10038161003816100381610038161003816⟨1x1003816100381610038161003816120601 (HΛ (120596))

1003816100381610038161003816 1y⟩10038161003816100381610038161003816] le 119862119890

minus119886 ln1+119888120588(xy) (98)










same 119888 119886 gt 0 as in (98)

E[ sup120601isinC1(119868)

10038161003816100381610038161003816⟨1x1003816100381610038161003816120601 (H (120596))

1003816100381610038161003816 1y⟩10038161003816100381610038161003816] le 119862eminus119886 ln

1+119888120588(xy) (99)




int120601 (120582) 119889120583xyB120596 (120582)

= ⟨1x1003816100381610038161003816 120601 (HB (120596))Π119868

(HB (120596))100381610038161003816100381610038161y⟩

(100)








119899rarr H with







119871119896

E [1003816100381610038161003816120583

xy120596


E [100381610038161003816100381610038161003816120583xyB119871119896

100381610038161003816100381610038161003816(E)] (101)




120601119905 120582 997891997888rarr 119891

119868(120582) ei119905120582 119905 isin R (102)








984858 = 2120575

120572 = 1 + 4120575

120591 =120575

2

(103)



120574 (119898 119871 119899) = 119898 (1 + 119871minus120591)119873minus119899+1

(104)



120588(x y) ge 1198621198731198711+120575 with 119862

119873= 4119873






119871(x) admits

a factorization

B(119873)

119871(x) = B(119899

1015840)

119871(x1015840) times B(119899

10158401015840)

119871(x10158401015840) (105)


119871(x) B


then 120588(ΠB119871(x) ΠB

119871(y)) ge 1198711+120575




119877119877 ge 119903



119880(2)

119877(119903) = 1

119903le119877119880

(2)(119903) (106)


gt



119877(x1015840) + U



11987110158401015840(x10158401015840) with

120588(B1198711015840(x1015840)B



H(119877)

B = H(119877)

B1198711015840(x1015840) otimes 1

(11989910158401015840)+ 1(119899

1015840)otimesH(119877)

B11987110158401015840

(x10158401015840) (107)


B(119873)


119873= 4119898119873


119871(x) =

B1198711015840(x1015840) times B

11987110158401015840(x10158401015840) with dist(x1015840 x10158401015840) gt (119862

1198732119873)119871 = 2119898119871

so that with 119877119896= (119862

1198732119873)119871

119896 one has

120598 (119877119896) lt eminus2119898119871119896 lt

1

2eminus119871120573

119896 (108)


119871119896(u) =

B(1198991015840)

119871119896(uJ)timesB

(11989910158401015840)



(a) dist(x1015840 x10158401015840) gt 119877119896= (119862

1198732119873)119871

119896

(b) B(1198991015840)

119871119896(u1015840) and B(119899

10158401015840)

119871119896(u10158401015840) are119898-Loc

(c) B(119873)

119871119896(u) is 119864-NR


119871119896(u) is (119864119898)-NS

Proof Set 119877119896= (119862

1198732119873)119871


above Denote

H(119877119896119873)

u119896 = H(119877119896)

B(119873)119871119896 (u)

G (119864) = (HB(119873)119871119896 (u)minus 119864)

minus1

(109)

and G(119877119896)(119864) = (H(119877119896119873)



H(119877119896 119873)

u119896 = HB(1198991015840)119871119896

(u1015840) otimes 1(11989910158401015840)+ 1(119899

1015840)otimesHB(119899

1015840)119871119896

(x10158401015840) (110)

with u = (u1015840 u10158401015840) isin Z119889119899

1015840

times Z11988911989910158401015840


119886119887= 120582

119886+120583

119887 where 120582

119886 = Σ(HB119871119896 (u



119887 = Σ(HB119871119896 (u


of H(119877119896 119873)


119886otimes 120595

119887 where


1015840)119871119896


of HB(11989910158401015840)119871119896


119886+ 120583

119887




|119864 minus (120582119886+ 120583

119887)| ge e119871

120573


119887)| ge e119871

120573

119896 and alsoas |(119864 minus 120583

119887) minus 120582

119886)| ge e119871

120573


G(119877119896) (u y 119864)

= sum

120582119886

sum

120583119887

120601119886(u1015840)120601

119886(y1015840)120595

119887(u10158401015840)120595

119887(y10158401015840)

(120582119886+ 120583

119887) minus 119864

= sum

120582119886

P1015840

119886(u1015840 y1015840)G(119877119896)

B119871119896 (u10158401015840)(u10158401015840 y10158401015840 119864 minus 120582

119886)

= sum

120583119887

P10158401015840

119887(u10158401015840 y10158401015840)G(119877119896)

B119871119896 (u1015840)(u1015840 y1015840 119864 minus 120583

119887)

(111)


B119871119896 (u1015840)(119864minus120583

119887) andG(119877119896)

B119871119896 (u10158401015840)(119864minus120582

119886) are

nonresonant1003817100381710038171003817100381710038171003817G(119877119896)

B119871119896 (u1015840)(119864 minus 120583

119887)

1003817100381710038171003817100381710038171003817le e119871

120573

119896

1003817100381710038171003817100381710038171003817G(119877119896)

B119871119896 (u10158401015840)(119864 minus 120582

119886)

1003817100381710038171003817100381710038171003817le e119871

120573

119896

(112)


y1015840) = 119871119896 in which case we

infer from (b)

10038161003816100381610038161003816G(119877119896) (u y 119864)10038161003816100381610038161003816 le

10038161003816100381610038161003816B119871119896(u10158401015840)10038161003816100381610038161003816eminus120574(119898119871119896119873minus1)119871119896+119871

120573

119896 (113)

or 120588(u10158401015840 y10158401015840) = 119871


10038161003816100381610038161003816G(119877119896) (u y 119864)10038161003816100381610038161003816 le

10038161003816100381610038161003816B119871119896(u1015840)10038161003816100381610038161003816eminus120574(119898119871119896119873minus1)119871119896+119871

120573

119896 (114)



119871119896+ 119871

120573

119896+ 119862 ln 119871

119896)

lt1

2eminus120574(119898119871119896119873)

(115)




10038171003817100381710038171003817G minus G(119877119896)

10038171003817100381710038171003817le10038171003817100381710038171003817U minus U(119877119896)

10038171003817100381710038171003817

10038171003817100381710038171003817G(119877119896)

10038171003817100381710038171003817G le eminus2119898119871119896e2119871

120573

119896

le1

2eminus120574(119898119871119896119873)

(117)


10038161003816100381610038161003816G (x y 119864) minus G(119877119896) (x y 119864)10038161003816100381610038161003816 le

1

2eminus120574(119898119871119896 119873)


119871119896+1(u)

(118)

Lemma 33 Let B119871119896+1(x)B



P B119871119896+1(x) B

119871119896+1(y) 119886119903119890 (119864119898) -119878

lt1

4119871minus119875(119873119896+1)

119896+1

(119)

Proof Let

R(2)= exist119864 B

119871119896+1(x) B

119871119896+1(y) 119886119903119890 119864-R

S(2)= B

119871119896+1(x) B119871119896+1

(y) 119886119903119890 (119864119898) -S (120)

ThenPS(2) le PR(2)

+PS(2)R(2)


119871119896+1(x) B

119871119896+1(y) must be 119864-NR


B(119873)

119871119896+1(x) = B(119899

1015840)

119871119896+1(x1015840) times B(119899

1015840)

119871119896+1(x10158401015840) (121)


119871119896+1(x10158401015840) are119898-Loc and

B119871119896+1(x) is (119864119898)-S then B




B(119873)

119871119896+1(y) = B(119899

1015840)

119871119896+1(y1015840) times B(119899

10158401015840)

119871119896+1(y10158401015840) (122)

If both B(1198991015840)119871119896+1(y1015840) and B(11989910158401015840)

119871119896+1(y10158401015840) are 119898-Loc and

B(119873)

119871119896+1(y) is (119864119898)-S then B(119873)

119871119896+1(y)must be 119864-R Let


1015840)

119871119896+1(x1015840) B(119899

10158401015840)

119871119896+1(x10158401015840)

B(1198991015840)

119871119896+1(y1015840) B(119899

1015840)

119871119896+1(y10158401015840) is 119898-NL119900119888

(123)


P L(4) le 4119871

minus119875(119873minus1119896+1)

119896+1= 4119871

minus2119875(119873119896+1)

119896+1 (124)



large enough

P S(2) le P R

(2) + P L

(4)

le eminus119871120573

119896 + 4119871minus2119875(119873119896+1)

119896+1lt1

4119871minus119875(119873119896+1)

119896+1

(125)






B119871119896(k) sub B

119871119896+1(u)with 120588(w k) ge 119862

1198731198711+120575

119896is (119864119898)-NS In the







119896also large)

1198711+984858

119896≫ 119871

1+120575


120588 (xB(119862119873minus1)119871119896

(w)) + 120588 (yB(119862119873minus1)119871119896(w))

ge 119877 minus (2119862119873minus 2) 119871

1+120575

119896

(126)

Assume first that

1199031015840= 120588 (xB

(119862119873minus1)1198711+120575119896(w)) ge 119871

119896+ 1

11990310158401015840= 120588 (yB

(119862119873minus1)1198711+120575119896(w)) ge 119871119896 + 1

(127)


1199031015840(x) and in B

11990310158401015840(y) are


1)1198711+120575

119896minus 2 ge 119877 minus 2119862

1198731198711+120575

119896










(119877minus21198621198731198711+120575119896 minus2119871119896)(119871119896+1)

sdot e119871120573

119896+1] = 119898(1 +1

2119871minus120591

119896)

119869119871119896

119871119896+ 1119877(1 minus

3119862119873

119877

sdot 119871119896) minus 119871

120573

119896+1= 119898119877[(1 +

1

2119871minus120591

119896)

119869


119896)


119873119871119896

1198711+984858

119896

) minus119871120573

119896+1

119898119877] = 119898119877[(1 +

1

2119871minus120591

119896)

119869


119896) (1 minus 3119862

119873119871minus984858

119896) minus

119871120573

119896+1

1198981198711+984858

119896

] ge 119898(1 +1

4


119896)

119869

119877 ge 120574 (119898 119871119896+1) 119877

(128)






1 119909

119873) 119909





1003816100381610038161003816 le 119862119889119871119889 (129)





by the functions

Φa =1

radic119873

sum

120587isinS119873

119873

⨂

119895=1

1119886120587minus1(119895) (130)

where a = 1198861 119886

119873 with 119886

1 119886

119873 = 119873



set is

ZNgt

= a = 1198861 119886

119873 119886

119895isinZ 119886

1 119886

119873 = 119873

(131)



1 119888

2 119888

119873 b = 119887

1 119888

2 119888

119873 with

1198861 119887

1 119888

2 119888

119873 = 119873 + 1

(ii) dZ(1198861 1198871) = 1


120588 (x y) = min120587isinS119873

max1le119895le119873

dZ (119909120587minus1(119895) 119910119895) (132)




119892V(x 120596) + U(x) where

U (x) = sum119894 =119895

119880(d (119909119894 119909

119895)) 119880 N 997888rarr R (133)


V (x 120596) = 119881 (1199091 120596) + sdot sdot sdot + 119881 (119909

119873 120596) (134)






Appendix



V (1199091 119909

119873 120596) = sum

119910isinxn119910119881 (119910 120596) (A1)





119908(y)) 119881 (119908 120596)

+ sum

V =119908

119888V119881 (V 120596) (A2)


119888V = nV (x) minus nV (y) (A3)



119908 while

n119908(x) minus n

119908(y) = 119888

119908isin Z 0 hence 1003816100381610038161003816119888119908

1003816100381610038161003816 ge 1 (A4)

Therefore

P 1003816100381610038161003816119892V (x 120596) minus 119892V (y 120596)

1003816100381610038161003816 le 119904

= E [P 1003816100381610038161003816V (x 120596) minus V (y 120596)

1003816100381610038161003816 le10038161003816100381610038161198921003816100381610038161003816

minus1119904 | F

=119908]

= E [P 1003816100381610038161003816119888119908119881 (119908 120596) | z (120596)

1003816100381610038161003816 le10038161003816100381610038161198921003816100381610038161003816

minus1119904 | F

=119908]

(A5)




120596 119888119908119881 (119908 120596) isin [z (120596) minus

10038161003816100381610038161198921003816100381610038161003816

minus1119904 z (120596) +

10038161003816100381610038161198921003816100381610038161003816

minus1119904]

= 120596 119881 (119908 120596)

isin [z (120596) minus11990410038161003816100381610038161198921003816100381610038161003816

minus1

119888119908

z (120596) +11990410038161003816100381610038161198921003816100381610038161003816

minus1

119888119908

]

(A6)


P 1003816100381610038161003816V (x 120596) minus V (y 120596)

1003816100381610038161003816 le10038161003816100381610038161198921003816100381610038161003816

minus1119904

le sup119886isinR

(119865119881119908(119886 + 2119904) minus 119865

119881119908(119886))

(A7)


at 119908 given F=119908 Since 119865




1003816100381610038161003816 le 119904 (A8)


1198710(u) admits



0 To prove the




100381610038161003816100381610038161003816GB1198710 (u)

(x y 119864)100381610038161003816100381610038161003816le eminus119898(119904)d(xy)

(A9)





119899

119895=11198612119871(119909

119895) If




1198711+120575(x) can be


(⋃

119895isinJ

1198612119871(119909

119895))∐(⋃

119894isinJ119888

1198612119871(119909

119894)) (A10)


120588 (119861119871(119909

119894) 119861

119871(119909

119895)) gt 2119871 (A11)


119873119871 then for some 119895

∘ 120588(119909

119895∘ 119910

119895∘) gt 119862

119873119871


120588 (119909119895 119909

119895∘) le diamΠx le 119860

119873119871

120588 (119910119895 119910

119895∘) le diamΠy lt 119860

119873119871

(A12)


120588 (119909119894 119910

119895) ge 120588 (119909

119895∘ 119910

119895∘) minus 120588 (119909

119894 119909

119895∘) minus 120588 (119910

119895 119910

119895∘)

ge 119862119873119871 minus 2119860

119873119871 ge 3119871

(A13)


yielding

120588 (Π119861119871(x) Π119861

119871(y)) gt 3119871 minus 2119871 = 119871 (A14)



Acknowledgments


References

































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










pair of 119898-NL119900119888 balls B(119873)

119871119896minus1(y1015840) B(119873)


(12)1198711+120575


(1) Both B(119873)

119871119896minus1(y1015840) and B(119873)

119871119896minus1(y10158401015840) are FI


120588 (ΠB119871119896minus1(u) ΠB

119871119896minus1(k)) ge 2119871

119896 (82)


119896minus1= 119871

11205722

119896+1and 119871

0large enough

P B(119873)

119871119896minus1(y1015840) B(119873)

119871119896minus1(y10158401015840) are 119898-NL119900119888

le 2119871minus(2119901120572

2)(1+120579)

119896minus1

119896+1

(83)


119871119896(w)|2 le (321198732)1198712119873


P S(2)

119896R

(2)

119896 le32119873

2119871minus(2119901120572

2)(1+120579)

119896minus1+2119873

119896+1 (84)



119896+1 under con-


2119901

12057222119873minus119873119901 (1 + 120579)

119896minus1minus 2119873119889 gt 2

119873minus119873119901 (1 + 120579)

119896+1 (85)

or equivalently

(1 + 120579)2lt2119901

1205722minus

2119873119889

119901 (1 + 120579)119896minus1 (86)


3120599 le2 minus 120572

2

1205722minus2119873

119901 (87)


P S(2)

119896 le P R

(2)

119896 + P S

(2)

119896R

(2)

119896

le eminus1198711205731015840

119896 +1

4119871minus119875(119873119896+1)

119896+1lt1

2119871minus119875(119873119896+1)

119896+1

(88)


119871119896minus1(y1015840) orB(119873)


P B(119873)

119871119896minus1(y1015840) B(119873)

119871119896minus1(y10158401015840) is 119898-NL119900119888

le min (P B(119873)

119871119896minus1(y1015840) is 119898-NL119900119888

P B(119873)

119871119896minus1(y10158401015840) is 119898-NL119900119888) le 119871

minus2119901(1+120579)119896minus1

119896minus1

lt 119871minus119901(1+120579)

119896+1

119896+1

(89)



119871minus119875(119873119896)

119896equiv 119871

minus119901(1+120599)119896


1+119888119871119896 (90)




P B(119873)

119871119896(u) 119894119904 119898-NL119900119888 le

1

4119871minus119901(1+120579)

119896

119896(91)


minus119901(1+120579)119896


119895of operator HB(119873)119871119896 (u)



119896one

has10038161003816100381610038161003816Ψ119895(x) Ψ






1003816100381610038161003816Ψ (y)1003816100381610038161003816 le eminus119898y

(93)








119871119896(y) sub

B119871119896+2(z) Fix 120596 isin Ω1015840


2le 1 implies Ψ

infinle 1 there is


119899isin B

119871119896minus1(0) then






119871119896+2(z) with 120588(y z) isin [119871

119896+1 119871

119896+2)


1+120575

119896) is (120582119898)-NS Note that



119877(y) with

119877 = 120588(y z)minus21198711+120575119896minus1 gt (12)119871

1+2120575



minusln 1003816100381610038161003816Ψ (y)

1003816100381610038161003816

120588 (y z)ge 119898 (1 + 119871

minus120591

119896) (1 minus

2119871119896+ 1

120588 (y z))

ge 119898(1 +1

2119871minus120591

119896) gt 119898

(95)









119871(119909) B


P exist 119864 isin 119868 B119871(x) B

119871(y) 119886119903119890 (119864119898) -S le 119891 (119871) (96)


119871(119909) cup B

119871(119910) one has

E[ sup120601isinB1(119868)

10038161003816100381610038161003816⟨1x1003816100381610038161003816120601 (HΛ (120596))

1003816100381610038161003816 1y⟩10038161003816100381610038161003816]

le 119862119871119889119890minus119898119871+ 119891 (119871)

(97)




Λ sup B119871(119909) cup B

119871(119910)


10038161003816100381610038161003816⟨1x1003816100381610038161003816120601 (HΛ (120596))

1003816100381610038161003816 1y⟩10038161003816100381610038161003816] le 119862119890

minus119886 ln1+119888120588(xy) (98)










same 119888 119886 gt 0 as in (98)

E[ sup120601isinC1(119868)

10038161003816100381610038161003816⟨1x1003816100381610038161003816120601 (H (120596))

1003816100381610038161003816 1y⟩10038161003816100381610038161003816] le 119862eminus119886 ln

1+119888120588(xy) (99)




int120601 (120582) 119889120583xyB120596 (120582)

= ⟨1x1003816100381610038161003816 120601 (HB (120596))Π119868

(HB (120596))100381610038161003816100381610038161y⟩

(100)








119899rarr H with







119871119896

E [1003816100381610038161003816120583

xy120596


E [100381610038161003816100381610038161003816120583xyB119871119896

100381610038161003816100381610038161003816(E)] (101)




120601119905 120582 997891997888rarr 119891

119868(120582) ei119905120582 119905 isin R (102)








984858 = 2120575

120572 = 1 + 4120575

120591 =120575

2

(103)



120574 (119898 119871 119899) = 119898 (1 + 119871minus120591)119873minus119899+1

(104)



120588(x y) ge 1198621198731198711+120575 with 119862

119873= 4119873






119871(x) admits

a factorization

B(119873)

119871(x) = B(119899

1015840)

119871(x1015840) times B(119899

10158401015840)

119871(x10158401015840) (105)


119871(x) B


then 120588(ΠB119871(x) ΠB

119871(y)) ge 1198711+120575




119877119877 ge 119903



119880(2)

119877(119903) = 1

119903le119877119880

(2)(119903) (106)


gt



119877(x1015840) + U



11987110158401015840(x10158401015840) with

120588(B1198711015840(x1015840)B



H(119877)

B = H(119877)

B1198711015840(x1015840) otimes 1

(11989910158401015840)+ 1(119899

1015840)otimesH(119877)

B11987110158401015840

(x10158401015840) (107)


B(119873)


119873= 4119898119873


119871(x) =

B1198711015840(x1015840) times B

11987110158401015840(x10158401015840) with dist(x1015840 x10158401015840) gt (119862

1198732119873)119871 = 2119898119871

so that with 119877119896= (119862

1198732119873)119871

119896 one has

120598 (119877119896) lt eminus2119898119871119896 lt

1

2eminus119871120573

119896 (108)


119871119896(u) =

B(1198991015840)

119871119896(uJ)timesB

(11989910158401015840)



(a) dist(x1015840 x10158401015840) gt 119877119896= (119862

1198732119873)119871

119896

(b) B(1198991015840)

119871119896(u1015840) and B(119899

10158401015840)

119871119896(u10158401015840) are119898-Loc

(c) B(119873)

119871119896(u) is 119864-NR


119871119896(u) is (119864119898)-NS

Proof Set 119877119896= (119862

1198732119873)119871


above Denote

H(119877119896119873)

u119896 = H(119877119896)

B(119873)119871119896 (u)

G (119864) = (HB(119873)119871119896 (u)minus 119864)

minus1

(109)

and G(119877119896)(119864) = (H(119877119896119873)



H(119877119896 119873)

u119896 = HB(1198991015840)119871119896

(u1015840) otimes 1(11989910158401015840)+ 1(119899

1015840)otimesHB(119899

1015840)119871119896

(x10158401015840) (110)

with u = (u1015840 u10158401015840) isin Z119889119899

1015840

times Z11988911989910158401015840


119886119887= 120582

119886+120583

119887 where 120582

119886 = Σ(HB119871119896 (u



119887 = Σ(HB119871119896 (u


of H(119877119896 119873)


119886otimes 120595

119887 where


1015840)119871119896


of HB(11989910158401015840)119871119896


119886+ 120583

119887




|119864 minus (120582119886+ 120583

119887)| ge e119871

120573


119887)| ge e119871

120573

119896 and alsoas |(119864 minus 120583

119887) minus 120582

119886)| ge e119871

120573


G(119877119896) (u y 119864)

= sum

120582119886

sum

120583119887

120601119886(u1015840)120601

119886(y1015840)120595

119887(u10158401015840)120595

119887(y10158401015840)

(120582119886+ 120583

119887) minus 119864

= sum

120582119886

P1015840

119886(u1015840 y1015840)G(119877119896)

B119871119896 (u10158401015840)(u10158401015840 y10158401015840 119864 minus 120582

119886)

= sum

120583119887

P10158401015840

119887(u10158401015840 y10158401015840)G(119877119896)

B119871119896 (u1015840)(u1015840 y1015840 119864 minus 120583

119887)

(111)


B119871119896 (u1015840)(119864minus120583

119887) andG(119877119896)

B119871119896 (u10158401015840)(119864minus120582

119886) are

nonresonant1003817100381710038171003817100381710038171003817G(119877119896)

B119871119896 (u1015840)(119864 minus 120583

119887)

1003817100381710038171003817100381710038171003817le e119871

120573

119896

1003817100381710038171003817100381710038171003817G(119877119896)

B119871119896 (u10158401015840)(119864 minus 120582

119886)

1003817100381710038171003817100381710038171003817le e119871

120573

119896

(112)


y1015840) = 119871119896 in which case we

infer from (b)

10038161003816100381610038161003816G(119877119896) (u y 119864)10038161003816100381610038161003816 le

10038161003816100381610038161003816B119871119896(u10158401015840)10038161003816100381610038161003816eminus120574(119898119871119896119873minus1)119871119896+119871

120573

119896 (113)

or 120588(u10158401015840 y10158401015840) = 119871


10038161003816100381610038161003816G(119877119896) (u y 119864)10038161003816100381610038161003816 le

10038161003816100381610038161003816B119871119896(u1015840)10038161003816100381610038161003816eminus120574(119898119871119896119873minus1)119871119896+119871

120573

119896 (114)



119871119896+ 119871

120573

119896+ 119862 ln 119871

119896)

lt1

2eminus120574(119898119871119896119873)

(115)




10038171003817100381710038171003817G minus G(119877119896)

10038171003817100381710038171003817le10038171003817100381710038171003817U minus U(119877119896)

10038171003817100381710038171003817

10038171003817100381710038171003817G(119877119896)

10038171003817100381710038171003817G le eminus2119898119871119896e2119871

120573

119896

le1

2eminus120574(119898119871119896119873)

(117)


10038161003816100381610038161003816G (x y 119864) minus G(119877119896) (x y 119864)10038161003816100381610038161003816 le

1

2eminus120574(119898119871119896 119873)


119871119896+1(u)

(118)

Lemma 33 Let B119871119896+1(x)B



P B119871119896+1(x) B

119871119896+1(y) 119886119903119890 (119864119898) -119878

lt1

4119871minus119875(119873119896+1)

119896+1

(119)

Proof Let

R(2)= exist119864 B

119871119896+1(x) B

119871119896+1(y) 119886119903119890 119864-R

S(2)= B

119871119896+1(x) B119871119896+1

(y) 119886119903119890 (119864119898) -S (120)

ThenPS(2) le PR(2)

+PS(2)R(2)


119871119896+1(x) B

119871119896+1(y) must be 119864-NR


B(119873)

119871119896+1(x) = B(119899

1015840)

119871119896+1(x1015840) times B(119899

1015840)

119871119896+1(x10158401015840) (121)


119871119896+1(x10158401015840) are119898-Loc and

B119871119896+1(x) is (119864119898)-S then B




B(119873)

119871119896+1(y) = B(119899

1015840)

119871119896+1(y1015840) times B(119899

10158401015840)

119871119896+1(y10158401015840) (122)

If both B(1198991015840)119871119896+1(y1015840) and B(11989910158401015840)

119871119896+1(y10158401015840) are 119898-Loc and

B(119873)

119871119896+1(y) is (119864119898)-S then B(119873)

119871119896+1(y)must be 119864-R Let


1015840)

119871119896+1(x1015840) B(119899

10158401015840)

119871119896+1(x10158401015840)

B(1198991015840)

119871119896+1(y1015840) B(119899

1015840)

119871119896+1(y10158401015840) is 119898-NL119900119888

(123)


P L(4) le 4119871

minus119875(119873minus1119896+1)

119896+1= 4119871

minus2119875(119873119896+1)

119896+1 (124)



large enough

P S(2) le P R

(2) + P L

(4)

le eminus119871120573

119896 + 4119871minus2119875(119873119896+1)

119896+1lt1

4119871minus119875(119873119896+1)

119896+1

(125)






B119871119896(k) sub B

119871119896+1(u)with 120588(w k) ge 119862

1198731198711+120575

119896is (119864119898)-NS In the







119896also large)

1198711+984858

119896≫ 119871

1+120575


120588 (xB(119862119873minus1)119871119896

(w)) + 120588 (yB(119862119873minus1)119871119896(w))

ge 119877 minus (2119862119873minus 2) 119871

1+120575

119896

(126)

Assume first that

1199031015840= 120588 (xB

(119862119873minus1)1198711+120575119896(w)) ge 119871

119896+ 1

11990310158401015840= 120588 (yB

(119862119873minus1)1198711+120575119896(w)) ge 119871119896 + 1

(127)


1199031015840(x) and in B

11990310158401015840(y) are


1)1198711+120575

119896minus 2 ge 119877 minus 2119862

1198731198711+120575

119896










(119877minus21198621198731198711+120575119896 minus2119871119896)(119871119896+1)

sdot e119871120573

119896+1] = 119898(1 +1

2119871minus120591

119896)

119869119871119896

119871119896+ 1119877(1 minus

3119862119873

119877

sdot 119871119896) minus 119871

120573

119896+1= 119898119877[(1 +

1

2119871minus120591

119896)

119869


119896)


119873119871119896

1198711+984858

119896

) minus119871120573

119896+1

119898119877] = 119898119877[(1 +

1

2119871minus120591

119896)

119869


119896) (1 minus 3119862

119873119871minus984858

119896) minus

119871120573

119896+1

1198981198711+984858

119896

] ge 119898(1 +1

4


119896)

119869

119877 ge 120574 (119898 119871119896+1) 119877

(128)






1 119909

119873) 119909





1003816100381610038161003816 le 119862119889119871119889 (129)





by the functions

Φa =1

radic119873

sum

120587isinS119873

119873

⨂

119895=1

1119886120587minus1(119895) (130)

where a = 1198861 119886

119873 with 119886

1 119886

119873 = 119873



set is

ZNgt

= a = 1198861 119886

119873 119886

119895isinZ 119886

1 119886

119873 = 119873

(131)



1 119888

2 119888

119873 b = 119887

1 119888

2 119888

119873 with

1198861 119887

1 119888

2 119888

119873 = 119873 + 1

(ii) dZ(1198861 1198871) = 1


120588 (x y) = min120587isinS119873

max1le119895le119873

dZ (119909120587minus1(119895) 119910119895) (132)




119892V(x 120596) + U(x) where

U (x) = sum119894 =119895

119880(d (119909119894 119909

119895)) 119880 N 997888rarr R (133)


V (x 120596) = 119881 (1199091 120596) + sdot sdot sdot + 119881 (119909

119873 120596) (134)






Appendix



V (1199091 119909

119873 120596) = sum

119910isinxn119910119881 (119910 120596) (A1)





119908(y)) 119881 (119908 120596)

+ sum

V =119908

119888V119881 (V 120596) (A2)


119888V = nV (x) minus nV (y) (A3)



119908 while

n119908(x) minus n

119908(y) = 119888

119908isin Z 0 hence 1003816100381610038161003816119888119908

1003816100381610038161003816 ge 1 (A4)

Therefore

P 1003816100381610038161003816119892V (x 120596) minus 119892V (y 120596)

1003816100381610038161003816 le 119904

= E [P 1003816100381610038161003816V (x 120596) minus V (y 120596)

1003816100381610038161003816 le10038161003816100381610038161198921003816100381610038161003816

minus1119904 | F

=119908]

= E [P 1003816100381610038161003816119888119908119881 (119908 120596) | z (120596)

1003816100381610038161003816 le10038161003816100381610038161198921003816100381610038161003816

minus1119904 | F

=119908]

(A5)




120596 119888119908119881 (119908 120596) isin [z (120596) minus

10038161003816100381610038161198921003816100381610038161003816

minus1119904 z (120596) +

10038161003816100381610038161198921003816100381610038161003816

minus1119904]

= 120596 119881 (119908 120596)

isin [z (120596) minus11990410038161003816100381610038161198921003816100381610038161003816

minus1

119888119908

z (120596) +11990410038161003816100381610038161198921003816100381610038161003816

minus1

119888119908

]

(A6)


P 1003816100381610038161003816V (x 120596) minus V (y 120596)

1003816100381610038161003816 le10038161003816100381610038161198921003816100381610038161003816

minus1119904

le sup119886isinR

(119865119881119908(119886 + 2119904) minus 119865

119881119908(119886))

(A7)


at 119908 given F=119908 Since 119865




1003816100381610038161003816 le 119904 (A8)


1198710(u) admits



0 To prove the




100381610038161003816100381610038161003816GB1198710 (u)

(x y 119864)100381610038161003816100381610038161003816le eminus119898(119904)d(xy)

(A9)





119899

119895=11198612119871(119909

119895) If




1198711+120575(x) can be


(⋃

119895isinJ

1198612119871(119909

119895))∐(⋃

119894isinJ119888

1198612119871(119909

119894)) (A10)


120588 (119861119871(119909

119894) 119861

119871(119909

119895)) gt 2119871 (A11)


119873119871 then for some 119895

∘ 120588(119909

119895∘ 119910

119895∘) gt 119862

119873119871


120588 (119909119895 119909

119895∘) le diamΠx le 119860

119873119871

120588 (119910119895 119910

119895∘) le diamΠy lt 119860

119873119871

(A12)


120588 (119909119894 119910

119895) ge 120588 (119909

119895∘ 119910

119895∘) minus 120588 (119909

119894 119909

119895∘) minus 120588 (119910

119895 119910

119895∘)

ge 119862119873119871 minus 2119860

119873119871 ge 3119871

(A13)


yielding

120588 (Π119861119871(x) Π119861

119871(y)) gt 3119871 minus 2119871 = 119871 (A14)



Acknowledgments


References

































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119871119896+2(z) with 120588(y z) isin [119871

119896+1 119871

119896+2)


1+120575

119896) is (120582119898)-NS Note that



119877(y) with

119877 = 120588(y z)minus21198711+120575119896minus1 gt (12)119871

1+2120575



minusln 1003816100381610038161003816Ψ (y)

1003816100381610038161003816

120588 (y z)ge 119898 (1 + 119871

minus120591

119896) (1 minus

2119871119896+ 1

120588 (y z))

ge 119898(1 +1

2119871minus120591

119896) gt 119898

(95)









119871(119909) B


P exist 119864 isin 119868 B119871(x) B

119871(y) 119886119903119890 (119864119898) -S le 119891 (119871) (96)


119871(119909) cup B

119871(119910) one has

E[ sup120601isinB1(119868)

10038161003816100381610038161003816⟨1x1003816100381610038161003816120601 (HΛ (120596))

1003816100381610038161003816 1y⟩10038161003816100381610038161003816]

le 119862119871119889119890minus119898119871+ 119891 (119871)

(97)




Λ sup B119871(119909) cup B

119871(119910)


10038161003816100381610038161003816⟨1x1003816100381610038161003816120601 (HΛ (120596))

1003816100381610038161003816 1y⟩10038161003816100381610038161003816] le 119862119890

minus119886 ln1+119888120588(xy) (98)










same 119888 119886 gt 0 as in (98)

E[ sup120601isinC1(119868)

10038161003816100381610038161003816⟨1x1003816100381610038161003816120601 (H (120596))

1003816100381610038161003816 1y⟩10038161003816100381610038161003816] le 119862eminus119886 ln

1+119888120588(xy) (99)




int120601 (120582) 119889120583xyB120596 (120582)

= ⟨1x1003816100381610038161003816 120601 (HB (120596))Π119868

(HB (120596))100381610038161003816100381610038161y⟩

(100)








119899rarr H with







119871119896

E [1003816100381610038161003816120583

xy120596


E [100381610038161003816100381610038161003816120583xyB119871119896

100381610038161003816100381610038161003816(E)] (101)




120601119905 120582 997891997888rarr 119891

119868(120582) ei119905120582 119905 isin R (102)








984858 = 2120575

120572 = 1 + 4120575

120591 =120575

2

(103)



120574 (119898 119871 119899) = 119898 (1 + 119871minus120591)119873minus119899+1

(104)



120588(x y) ge 1198621198731198711+120575 with 119862

119873= 4119873






119871(x) admits

a factorization

B(119873)

119871(x) = B(119899

1015840)

119871(x1015840) times B(119899

10158401015840)

119871(x10158401015840) (105)


119871(x) B


then 120588(ΠB119871(x) ΠB

119871(y)) ge 1198711+120575




119877119877 ge 119903



119880(2)

119877(119903) = 1

119903le119877119880

(2)(119903) (106)


gt



119877(x1015840) + U



11987110158401015840(x10158401015840) with

120588(B1198711015840(x1015840)B



H(119877)

B = H(119877)

B1198711015840(x1015840) otimes 1

(11989910158401015840)+ 1(119899

1015840)otimesH(119877)

B11987110158401015840

(x10158401015840) (107)


B(119873)


119873= 4119898119873


119871(x) =

B1198711015840(x1015840) times B

11987110158401015840(x10158401015840) with dist(x1015840 x10158401015840) gt (119862

1198732119873)119871 = 2119898119871

so that with 119877119896= (119862

1198732119873)119871

119896 one has

120598 (119877119896) lt eminus2119898119871119896 lt

1

2eminus119871120573

119896 (108)


119871119896(u) =

B(1198991015840)

119871119896(uJ)timesB

(11989910158401015840)



(a) dist(x1015840 x10158401015840) gt 119877119896= (119862

1198732119873)119871

119896

(b) B(1198991015840)

119871119896(u1015840) and B(119899

10158401015840)

119871119896(u10158401015840) are119898-Loc

(c) B(119873)

119871119896(u) is 119864-NR


119871119896(u) is (119864119898)-NS

Proof Set 119877119896= (119862

1198732119873)119871


above Denote

H(119877119896119873)

u119896 = H(119877119896)

B(119873)119871119896 (u)

G (119864) = (HB(119873)119871119896 (u)minus 119864)

minus1

(109)

and G(119877119896)(119864) = (H(119877119896119873)



H(119877119896 119873)

u119896 = HB(1198991015840)119871119896

(u1015840) otimes 1(11989910158401015840)+ 1(119899

1015840)otimesHB(119899

1015840)119871119896

(x10158401015840) (110)

with u = (u1015840 u10158401015840) isin Z119889119899

1015840

times Z11988911989910158401015840


119886119887= 120582

119886+120583

119887 where 120582

119886 = Σ(HB119871119896 (u



119887 = Σ(HB119871119896 (u


of H(119877119896 119873)


119886otimes 120595

119887 where


1015840)119871119896


of HB(11989910158401015840)119871119896


119886+ 120583

119887




|119864 minus (120582119886+ 120583

119887)| ge e119871

120573


119887)| ge e119871

120573

119896 and alsoas |(119864 minus 120583

119887) minus 120582

119886)| ge e119871

120573


G(119877119896) (u y 119864)

= sum

120582119886

sum

120583119887

120601119886(u1015840)120601

119886(y1015840)120595

119887(u10158401015840)120595

119887(y10158401015840)

(120582119886+ 120583

119887) minus 119864

= sum

120582119886

P1015840

119886(u1015840 y1015840)G(119877119896)

B119871119896 (u10158401015840)(u10158401015840 y10158401015840 119864 minus 120582

119886)

= sum

120583119887

P10158401015840

119887(u10158401015840 y10158401015840)G(119877119896)

B119871119896 (u1015840)(u1015840 y1015840 119864 minus 120583

119887)

(111)


B119871119896 (u1015840)(119864minus120583

119887) andG(119877119896)

B119871119896 (u10158401015840)(119864minus120582

119886) are

nonresonant1003817100381710038171003817100381710038171003817G(119877119896)

B119871119896 (u1015840)(119864 minus 120583

119887)

1003817100381710038171003817100381710038171003817le e119871

120573

119896

1003817100381710038171003817100381710038171003817G(119877119896)

B119871119896 (u10158401015840)(119864 minus 120582

119886)

1003817100381710038171003817100381710038171003817le e119871

120573

119896

(112)


y1015840) = 119871119896 in which case we

infer from (b)

10038161003816100381610038161003816G(119877119896) (u y 119864)10038161003816100381610038161003816 le

10038161003816100381610038161003816B119871119896(u10158401015840)10038161003816100381610038161003816eminus120574(119898119871119896119873minus1)119871119896+119871

120573

119896 (113)

or 120588(u10158401015840 y10158401015840) = 119871


10038161003816100381610038161003816G(119877119896) (u y 119864)10038161003816100381610038161003816 le

10038161003816100381610038161003816B119871119896(u1015840)10038161003816100381610038161003816eminus120574(119898119871119896119873minus1)119871119896+119871

120573

119896 (114)



119871119896+ 119871

120573

119896+ 119862 ln 119871

119896)

lt1

2eminus120574(119898119871119896119873)

(115)




10038171003817100381710038171003817G minus G(119877119896)

10038171003817100381710038171003817le10038171003817100381710038171003817U minus U(119877119896)

10038171003817100381710038171003817

10038171003817100381710038171003817G(119877119896)

10038171003817100381710038171003817G le eminus2119898119871119896e2119871

120573

119896

le1

2eminus120574(119898119871119896119873)

(117)


10038161003816100381610038161003816G (x y 119864) minus G(119877119896) (x y 119864)10038161003816100381610038161003816 le

1

2eminus120574(119898119871119896 119873)


119871119896+1(u)

(118)

Lemma 33 Let B119871119896+1(x)B



P B119871119896+1(x) B

119871119896+1(y) 119886119903119890 (119864119898) -119878

lt1

4119871minus119875(119873119896+1)

119896+1

(119)

Proof Let

R(2)= exist119864 B

119871119896+1(x) B

119871119896+1(y) 119886119903119890 119864-R

S(2)= B

119871119896+1(x) B119871119896+1

(y) 119886119903119890 (119864119898) -S (120)

ThenPS(2) le PR(2)

+PS(2)R(2)


119871119896+1(x) B

119871119896+1(y) must be 119864-NR


B(119873)

119871119896+1(x) = B(119899

1015840)

119871119896+1(x1015840) times B(119899

1015840)

119871119896+1(x10158401015840) (121)


119871119896+1(x10158401015840) are119898-Loc and

B119871119896+1(x) is (119864119898)-S then B




B(119873)

119871119896+1(y) = B(119899

1015840)

119871119896+1(y1015840) times B(119899

10158401015840)

119871119896+1(y10158401015840) (122)

If both B(1198991015840)119871119896+1(y1015840) and B(11989910158401015840)

119871119896+1(y10158401015840) are 119898-Loc and

B(119873)

119871119896+1(y) is (119864119898)-S then B(119873)

119871119896+1(y)must be 119864-R Let


1015840)

119871119896+1(x1015840) B(119899

10158401015840)

119871119896+1(x10158401015840)

B(1198991015840)

119871119896+1(y1015840) B(119899

1015840)

119871119896+1(y10158401015840) is 119898-NL119900119888

(123)


P L(4) le 4119871

minus119875(119873minus1119896+1)

119896+1= 4119871

minus2119875(119873119896+1)

119896+1 (124)



large enough

P S(2) le P R

(2) + P L

(4)

le eminus119871120573

119896 + 4119871minus2119875(119873119896+1)

119896+1lt1

4119871minus119875(119873119896+1)

119896+1

(125)






B119871119896(k) sub B

119871119896+1(u)with 120588(w k) ge 119862

1198731198711+120575

119896is (119864119898)-NS In the







119896also large)

1198711+984858

119896≫ 119871

1+120575


120588 (xB(119862119873minus1)119871119896

(w)) + 120588 (yB(119862119873minus1)119871119896(w))

ge 119877 minus (2119862119873minus 2) 119871

1+120575

119896

(126)

Assume first that

1199031015840= 120588 (xB

(119862119873minus1)1198711+120575119896(w)) ge 119871

119896+ 1

11990310158401015840= 120588 (yB

(119862119873minus1)1198711+120575119896(w)) ge 119871119896 + 1

(127)


1199031015840(x) and in B

11990310158401015840(y) are


1)1198711+120575

119896minus 2 ge 119877 minus 2119862

1198731198711+120575

119896










(119877minus21198621198731198711+120575119896 minus2119871119896)(119871119896+1)

sdot e119871120573

119896+1] = 119898(1 +1

2119871minus120591

119896)

119869119871119896

119871119896+ 1119877(1 minus

3119862119873

119877

sdot 119871119896) minus 119871

120573

119896+1= 119898119877[(1 +

1

2119871minus120591

119896)

119869


119896)


119873119871119896

1198711+984858

119896

) minus119871120573

119896+1

119898119877] = 119898119877[(1 +

1

2119871minus120591

119896)

119869


119896) (1 minus 3119862

119873119871minus984858

119896) minus

119871120573

119896+1

1198981198711+984858

119896

] ge 119898(1 +1

4


119896)

119869

119877 ge 120574 (119898 119871119896+1) 119877

(128)






1 119909

119873) 119909





1003816100381610038161003816 le 119862119889119871119889 (129)





by the functions

Φa =1

radic119873

sum

120587isinS119873

119873

⨂

119895=1

1119886120587minus1(119895) (130)

where a = 1198861 119886

119873 with 119886

1 119886

119873 = 119873



set is

ZNgt

= a = 1198861 119886

119873 119886

119895isinZ 119886

1 119886

119873 = 119873

(131)



1 119888

2 119888

119873 b = 119887

1 119888

2 119888

119873 with

1198861 119887

1 119888

2 119888

119873 = 119873 + 1

(ii) dZ(1198861 1198871) = 1


120588 (x y) = min120587isinS119873

max1le119895le119873

dZ (119909120587minus1(119895) 119910119895) (132)




119892V(x 120596) + U(x) where

U (x) = sum119894 =119895

119880(d (119909119894 119909

119895)) 119880 N 997888rarr R (133)


V (x 120596) = 119881 (1199091 120596) + sdot sdot sdot + 119881 (119909

119873 120596) (134)






Appendix



V (1199091 119909

119873 120596) = sum

119910isinxn119910119881 (119910 120596) (A1)





119908(y)) 119881 (119908 120596)

+ sum

V =119908

119888V119881 (V 120596) (A2)


119888V = nV (x) minus nV (y) (A3)



119908 while

n119908(x) minus n

119908(y) = 119888

119908isin Z 0 hence 1003816100381610038161003816119888119908

1003816100381610038161003816 ge 1 (A4)

Therefore

P 1003816100381610038161003816119892V (x 120596) minus 119892V (y 120596)

1003816100381610038161003816 le 119904

= E [P 1003816100381610038161003816V (x 120596) minus V (y 120596)

1003816100381610038161003816 le10038161003816100381610038161198921003816100381610038161003816

minus1119904 | F

=119908]

= E [P 1003816100381610038161003816119888119908119881 (119908 120596) | z (120596)

1003816100381610038161003816 le10038161003816100381610038161198921003816100381610038161003816

minus1119904 | F

=119908]

(A5)




120596 119888119908119881 (119908 120596) isin [z (120596) minus

10038161003816100381610038161198921003816100381610038161003816

minus1119904 z (120596) +

10038161003816100381610038161198921003816100381610038161003816

minus1119904]

= 120596 119881 (119908 120596)

isin [z (120596) minus11990410038161003816100381610038161198921003816100381610038161003816

minus1

119888119908

z (120596) +11990410038161003816100381610038161198921003816100381610038161003816

minus1

119888119908

]

(A6)


P 1003816100381610038161003816V (x 120596) minus V (y 120596)

1003816100381610038161003816 le10038161003816100381610038161198921003816100381610038161003816

minus1119904

le sup119886isinR

(119865119881119908(119886 + 2119904) minus 119865

119881119908(119886))

(A7)


at 119908 given F=119908 Since 119865




1003816100381610038161003816 le 119904 (A8)


1198710(u) admits



0 To prove the




100381610038161003816100381610038161003816GB1198710 (u)

(x y 119864)100381610038161003816100381610038161003816le eminus119898(119904)d(xy)

(A9)





119899

119895=11198612119871(119909

119895) If




1198711+120575(x) can be


(⋃

119895isinJ

1198612119871(119909

119895))∐(⋃

119894isinJ119888

1198612119871(119909

119894)) (A10)


120588 (119861119871(119909

119894) 119861

119871(119909

119895)) gt 2119871 (A11)


119873119871 then for some 119895

∘ 120588(119909

119895∘ 119910

119895∘) gt 119862

119873119871


120588 (119909119895 119909

119895∘) le diamΠx le 119860

119873119871

120588 (119910119895 119910

119895∘) le diamΠy lt 119860

119873119871

(A12)


120588 (119909119894 119910

119895) ge 120588 (119909

119895∘ 119910

119895∘) minus 120588 (119909

119894 119909

119895∘) minus 120588 (119910

119895 119910

119895∘)

ge 119862119873119871 minus 2119860

119873119871 ge 3119871

(A13)


yielding

120588 (Π119861119871(x) Π119861

119871(y)) gt 3119871 minus 2119871 = 119871 (A14)



Acknowledgments


References

































Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















984858 = 2120575

120572 = 1 + 4120575

120591 =120575

2

(103)



120574 (119898 119871 119899) = 119898 (1 + 119871minus120591)119873minus119899+1

(104)



120588(x y) ge 1198621198731198711+120575 with 119862

119873= 4119873






119871(x) admits

a factorization

B(119873)

119871(x) = B(119899

1015840)

119871(x1015840) times B(119899

10158401015840)

119871(x10158401015840) (105)


119871(x) B


then 120588(ΠB119871(x) ΠB

119871(y)) ge 1198711+120575




119877119877 ge 119903



119880(2)

119877(119903) = 1

119903le119877119880

(2)(119903) (106)


gt



119877(x1015840) + U



11987110158401015840(x10158401015840) with

120588(B1198711015840(x1015840)B



H(119877)

B = H(119877)

B1198711015840(x1015840) otimes 1

(11989910158401015840)+ 1(119899

1015840)otimesH(119877)

B11987110158401015840

(x10158401015840) (107)


B(119873)


119873= 4119898119873


119871(x) =

B1198711015840(x1015840) times B

11987110158401015840(x10158401015840) with dist(x1015840 x10158401015840) gt (119862

1198732119873)119871 = 2119898119871

so that with 119877119896= (119862

1198732119873)119871

119896 one has

120598 (119877119896) lt eminus2119898119871119896 lt

1

2eminus119871120573

119896 (108)


119871119896(u) =

B(1198991015840)

119871119896(uJ)timesB

(11989910158401015840)



(a) dist(x1015840 x10158401015840) gt 119877119896= (119862

1198732119873)119871

119896

(b) B(1198991015840)

119871119896(u1015840) and B(119899

10158401015840)

119871119896(u10158401015840) are119898-Loc

(c) B(119873)

119871119896(u) is 119864-NR


119871119896(u) is (119864119898)-NS

Proof Set 119877119896= (119862

1198732119873)119871


above Denote

H(119877119896119873)

u119896 = H(119877119896)

B(119873)119871119896 (u)

G (119864) = (HB(119873)119871119896 (u)minus 119864)

minus1

(109)

and G(119877119896)(119864) = (H(119877119896119873)



H(119877119896 119873)

u119896 = HB(1198991015840)119871119896

(u1015840) otimes 1(11989910158401015840)+ 1(119899

1015840)otimesHB(119899

1015840)119871119896

(x10158401015840) (110)

with u = (u1015840 u10158401015840) isin Z119889119899

1015840

times Z11988911989910158401015840


119886119887= 120582

119886+120583

119887 where 120582

119886 = Σ(HB119871119896 (u



119887 = Σ(HB119871119896 (u


of H(119877119896 119873)


119886otimes 120595

119887 where


1015840)119871119896


of HB(11989910158401015840)119871119896


119886+ 120583

119887




|119864 minus (120582119886+ 120583

119887)| ge e119871

120573


119887)| ge e119871

120573

119896 and alsoas |(119864 minus 120583

119887) minus 120582

119886)| ge e119871

120573


G(119877119896) (u y 119864)

= sum

120582119886

sum

120583119887

120601119886(u1015840)120601

119886(y1015840)120595

119887(u10158401015840)120595

119887(y10158401015840)

(120582119886+ 120583

119887) minus 119864

= sum

120582119886

P1015840

119886(u1015840 y1015840)G(119877119896)

B119871119896 (u10158401015840)(u10158401015840 y10158401015840 119864 minus 120582

119886)

= sum

120583119887

P10158401015840

119887(u10158401015840 y10158401015840)G(119877119896)

B119871119896 (u1015840)(u1015840 y1015840 119864 minus 120583

119887)

(111)


B119871119896 (u1015840)(119864minus120583

119887) andG(119877119896)

B119871119896 (u10158401015840)(119864minus120582

119886) are

nonresonant1003817100381710038171003817100381710038171003817G(119877119896)

B119871119896 (u1015840)(119864 minus 120583

119887)

1003817100381710038171003817100381710038171003817le e119871

120573

119896

1003817100381710038171003817100381710038171003817G(119877119896)

B119871119896 (u10158401015840)(119864 minus 120582

119886)

1003817100381710038171003817100381710038171003817le e119871

120573

119896

(112)


y1015840) = 119871119896 in which case we

infer from (b)

10038161003816100381610038161003816G(119877119896) (u y 119864)10038161003816100381610038161003816 le

10038161003816100381610038161003816B119871119896(u10158401015840)10038161003816100381610038161003816eminus120574(119898119871119896119873minus1)119871119896+119871

120573

119896 (113)

or 120588(u10158401015840 y10158401015840) = 119871


10038161003816100381610038161003816G(119877119896) (u y 119864)10038161003816100381610038161003816 le

10038161003816100381610038161003816B119871119896(u1015840)10038161003816100381610038161003816eminus120574(119898119871119896119873minus1)119871119896+119871

120573

119896 (114)



119871119896+ 119871

120573

119896+ 119862 ln 119871

119896)

lt1

2eminus120574(119898119871119896119873)

(115)




10038171003817100381710038171003817G minus G(119877119896)

10038171003817100381710038171003817le10038171003817100381710038171003817U minus U(119877119896)

10038171003817100381710038171003817

10038171003817100381710038171003817G(119877119896)

10038171003817100381710038171003817G le eminus2119898119871119896e2119871

120573

119896

le1

2eminus120574(119898119871119896119873)

(117)


10038161003816100381610038161003816G (x y 119864) minus G(119877119896) (x y 119864)10038161003816100381610038161003816 le

1

2eminus120574(119898119871119896 119873)


119871119896+1(u)

(118)

Lemma 33 Let B119871119896+1(x)B



P B119871119896+1(x) B

119871119896+1(y) 119886119903119890 (119864119898) -119878

lt1

4119871minus119875(119873119896+1)

119896+1

(119)

Proof Let

R(2)= exist119864 B

119871119896+1(x) B

119871119896+1(y) 119886119903119890 119864-R

S(2)= B

119871119896+1(x) B119871119896+1

(y) 119886119903119890 (119864119898) -S (120)

ThenPS(2) le PR(2)

+PS(2)R(2)


119871119896+1(x) B

119871119896+1(y) must be 119864-NR


B(119873)

119871119896+1(x) = B(119899

1015840)

119871119896+1(x1015840) times B(119899

1015840)

119871119896+1(x10158401015840) (121)


119871119896+1(x10158401015840) are119898-Loc and

B119871119896+1(x) is (119864119898)-S then B




B(119873)

119871119896+1(y) = B(119899

1015840)

119871119896+1(y1015840) times B(119899

10158401015840)

119871119896+1(y10158401015840) (122)

If both B(1198991015840)119871119896+1(y1015840) and B(11989910158401015840)

119871119896+1(y10158401015840) are 119898-Loc and

B(119873)

119871119896+1(y) is (119864119898)-S then B(119873)

119871119896+1(y)must be 119864-R Let


1015840)

119871119896+1(x1015840) B(119899

10158401015840)

119871119896+1(x10158401015840)

B(1198991015840)

119871119896+1(y1015840) B(119899

1015840)

119871119896+1(y10158401015840) is 119898-NL119900119888

(123)


P L(4) le 4119871

minus119875(119873minus1119896+1)

119896+1= 4119871

minus2119875(119873119896+1)

119896+1 (124)



large enough

P S(2) le P R

(2) + P L

(4)

le eminus119871120573

119896 + 4119871minus2119875(119873119896+1)

119896+1lt1

4119871minus119875(119873119896+1)

119896+1

(125)






B119871119896(k) sub B

119871119896+1(u)with 120588(w k) ge 119862

1198731198711+120575

119896is (119864119898)-NS In the







119896also large)

1198711+984858

119896≫ 119871

1+120575


120588 (xB(119862119873minus1)119871119896

(w)) + 120588 (yB(119862119873minus1)119871119896(w))

ge 119877 minus (2119862119873minus 2) 119871

1+120575

119896

(126)

Assume first that

1199031015840= 120588 (xB

(119862119873minus1)1198711+120575119896(w)) ge 119871

119896+ 1

11990310158401015840= 120588 (yB

(119862119873minus1)1198711+120575119896(w)) ge 119871119896 + 1

(127)


1199031015840(x) and in B

11990310158401015840(y) are


1)1198711+120575

119896minus 2 ge 119877 minus 2119862

1198731198711+120575

119896










(119877minus21198621198731198711+120575119896 minus2119871119896)(119871119896+1)

sdot e119871120573

119896+1] = 119898(1 +1

2119871minus120591

119896)

119869119871119896

119871119896+ 1119877(1 minus

3119862119873

119877

sdot 119871119896) minus 119871

120573

119896+1= 119898119877[(1 +

1

2119871minus120591

119896)

119869


119896)


119873119871119896

1198711+984858

119896

) minus119871120573

119896+1

119898119877] = 119898119877[(1 +

1

2119871minus120591

119896)

119869


119896) (1 minus 3119862

119873119871minus984858

119896) minus

119871120573

119896+1

1198981198711+984858

119896

] ge 119898(1 +1

4


119896)

119869

119877 ge 120574 (119898 119871119896+1) 119877

(128)






1 119909

119873) 119909





1003816100381610038161003816 le 119862119889119871119889 (129)





by the functions

Φa =1

radic119873

sum

120587isinS119873

119873

⨂

119895=1

1119886120587minus1(119895) (130)

where a = 1198861 119886

119873 with 119886

1 119886

119873 = 119873



set is

ZNgt

= a = 1198861 119886

119873 119886

119895isinZ 119886

1 119886

119873 = 119873

(131)



1 119888

2 119888

119873 b = 119887

1 119888

2 119888

119873 with

1198861 119887

1 119888

2 119888

119873 = 119873 + 1

(ii) dZ(1198861 1198871) = 1


120588 (x y) = min120587isinS119873

max1le119895le119873

dZ (119909120587minus1(119895) 119910119895) (132)




119892V(x 120596) + U(x) where

U (x) = sum119894 =119895

119880(d (119909119894 119909

119895)) 119880 N 997888rarr R (133)


V (x 120596) = 119881 (1199091 120596) + sdot sdot sdot + 119881 (119909

119873 120596) (134)






Appendix



V (1199091 119909

119873 120596) = sum

119910isinxn119910119881 (119910 120596) (A1)





119908(y)) 119881 (119908 120596)

+ sum

V =119908

119888V119881 (V 120596) (A2)


119888V = nV (x) minus nV (y) (A3)



119908 while

n119908(x) minus n

119908(y) = 119888

119908isin Z 0 hence 1003816100381610038161003816119888119908

1003816100381610038161003816 ge 1 (A4)

Therefore

P 1003816100381610038161003816119892V (x 120596) minus 119892V (y 120596)

1003816100381610038161003816 le 119904

= E [P 1003816100381610038161003816V (x 120596) minus V (y 120596)

1003816100381610038161003816 le10038161003816100381610038161198921003816100381610038161003816

minus1119904 | F

=119908]

= E [P 1003816100381610038161003816119888119908119881 (119908 120596) | z (120596)

1003816100381610038161003816 le10038161003816100381610038161198921003816100381610038161003816

minus1119904 | F

=119908]

(A5)




120596 119888119908119881 (119908 120596) isin [z (120596) minus

10038161003816100381610038161198921003816100381610038161003816

minus1119904 z (120596) +

10038161003816100381610038161198921003816100381610038161003816

minus1119904]

= 120596 119881 (119908 120596)

isin [z (120596) minus11990410038161003816100381610038161198921003816100381610038161003816

minus1

119888119908

z (120596) +11990410038161003816100381610038161198921003816100381610038161003816

minus1

119888119908

]

(A6)


P 1003816100381610038161003816V (x 120596) minus V (y 120596)

1003816100381610038161003816 le10038161003816100381610038161198921003816100381610038161003816

minus1119904

le sup119886isinR

(119865119881119908(119886 + 2119904) minus 119865

119881119908(119886))

(A7)


at 119908 given F=119908 Since 119865




1003816100381610038161003816 le 119904 (A8)


1198710(u) admits



0 To prove the




100381610038161003816100381610038161003816GB1198710 (u)

(x y 119864)100381610038161003816100381610038161003816le eminus119898(119904)d(xy)

(A9)





119899

119895=11198612119871(119909

119895) If




1198711+120575(x) can be


(⋃

119895isinJ

1198612119871(119909

119895))∐(⋃

119894isinJ119888

1198612119871(119909

119894)) (A10)


120588 (119861119871(119909

119894) 119861

119871(119909

119895)) gt 2119871 (A11)


119873119871 then for some 119895

∘ 120588(119909

119895∘ 119910

119895∘) gt 119862

119873119871


120588 (119909119895 119909

119895∘) le diamΠx le 119860

119873119871

120588 (119910119895 119910

119895∘) le diamΠy lt 119860

119873119871

(A12)


120588 (119909119894 119910

119895) ge 120588 (119909

119895∘ 119910

119895∘) minus 120588 (119909

119894 119909

119895∘) minus 120588 (119910

119895 119910

119895∘)

ge 119862119873119871 minus 2119860

119873119871 ge 3119871

(A13)


yielding

120588 (Π119861119871(x) Π119861

119871(y)) gt 3119871 minus 2119871 = 119871 (A14)



Acknowledgments


References

































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










|119864 minus (120582119886+ 120583

119887)| ge e119871

120573


119887)| ge e119871

120573

119896 and alsoas |(119864 minus 120583

119887) minus 120582

119886)| ge e119871

120573


G(119877119896) (u y 119864)

= sum

120582119886

sum

120583119887

120601119886(u1015840)120601

119886(y1015840)120595

119887(u10158401015840)120595

119887(y10158401015840)

(120582119886+ 120583

119887) minus 119864

= sum

120582119886

P1015840

119886(u1015840 y1015840)G(119877119896)

B119871119896 (u10158401015840)(u10158401015840 y10158401015840 119864 minus 120582

119886)

= sum

120583119887

P10158401015840

119887(u10158401015840 y10158401015840)G(119877119896)

B119871119896 (u1015840)(u1015840 y1015840 119864 minus 120583

119887)

(111)


B119871119896 (u1015840)(119864minus120583

119887) andG(119877119896)

B119871119896 (u10158401015840)(119864minus120582

119886) are

nonresonant1003817100381710038171003817100381710038171003817G(119877119896)

B119871119896 (u1015840)(119864 minus 120583

119887)

1003817100381710038171003817100381710038171003817le e119871

120573

119896

1003817100381710038171003817100381710038171003817G(119877119896)

B119871119896 (u10158401015840)(119864 minus 120582

119886)

1003817100381710038171003817100381710038171003817le e119871

120573

119896

(112)


y1015840) = 119871119896 in which case we

infer from (b)

10038161003816100381610038161003816G(119877119896) (u y 119864)10038161003816100381610038161003816 le

10038161003816100381610038161003816B119871119896(u10158401015840)10038161003816100381610038161003816eminus120574(119898119871119896119873minus1)119871119896+119871

120573

119896 (113)

or 120588(u10158401015840 y10158401015840) = 119871


10038161003816100381610038161003816G(119877119896) (u y 119864)10038161003816100381610038161003816 le

10038161003816100381610038161003816B119871119896(u1015840)10038161003816100381610038161003816eminus120574(119898119871119896119873minus1)119871119896+119871

120573

119896 (114)



119871119896+ 119871

120573

119896+ 119862 ln 119871

119896)

lt1

2eminus120574(119898119871119896119873)

(115)




10038171003817100381710038171003817G minus G(119877119896)

10038171003817100381710038171003817le10038171003817100381710038171003817U minus U(119877119896)

10038171003817100381710038171003817

10038171003817100381710038171003817G(119877119896)

10038171003817100381710038171003817G le eminus2119898119871119896e2119871

120573

119896

le1

2eminus120574(119898119871119896119873)

(117)


10038161003816100381610038161003816G (x y 119864) minus G(119877119896) (x y 119864)10038161003816100381610038161003816 le

1

2eminus120574(119898119871119896 119873)


119871119896+1(u)

(118)

Lemma 33 Let B119871119896+1(x)B



P B119871119896+1(x) B

119871119896+1(y) 119886119903119890 (119864119898) -119878

lt1

4119871minus119875(119873119896+1)

119896+1

(119)

Proof Let

R(2)= exist119864 B

119871119896+1(x) B

119871119896+1(y) 119886119903119890 119864-R

S(2)= B

119871119896+1(x) B119871119896+1

(y) 119886119903119890 (119864119898) -S (120)

ThenPS(2) le PR(2)

+PS(2)R(2)


119871119896+1(x) B

119871119896+1(y) must be 119864-NR


B(119873)

119871119896+1(x) = B(119899

1015840)

119871119896+1(x1015840) times B(119899

1015840)

119871119896+1(x10158401015840) (121)


119871119896+1(x10158401015840) are119898-Loc and

B119871119896+1(x) is (119864119898)-S then B




B(119873)

119871119896+1(y) = B(119899

1015840)

119871119896+1(y1015840) times B(119899

10158401015840)

119871119896+1(y10158401015840) (122)

If both B(1198991015840)119871119896+1(y1015840) and B(11989910158401015840)

119871119896+1(y10158401015840) are 119898-Loc and

B(119873)

119871119896+1(y) is (119864119898)-S then B(119873)

119871119896+1(y)must be 119864-R Let


1015840)

119871119896+1(x1015840) B(119899

10158401015840)

119871119896+1(x10158401015840)

B(1198991015840)

119871119896+1(y1015840) B(119899

1015840)

119871119896+1(y10158401015840) is 119898-NL119900119888

(123)


P L(4) le 4119871

minus119875(119873minus1119896+1)

119896+1= 4119871

minus2119875(119873119896+1)

119896+1 (124)



large enough

P S(2) le P R

(2) + P L

(4)

le eminus119871120573

119896 + 4119871minus2119875(119873119896+1)

119896+1lt1

4119871minus119875(119873119896+1)

119896+1

(125)






B119871119896(k) sub B

119871119896+1(u)with 120588(w k) ge 119862

1198731198711+120575

119896is (119864119898)-NS In the







119896also large)

1198711+984858

119896≫ 119871

1+120575


120588 (xB(119862119873minus1)119871119896

(w)) + 120588 (yB(119862119873minus1)119871119896(w))

ge 119877 minus (2119862119873minus 2) 119871

1+120575

119896

(126)

Assume first that

1199031015840= 120588 (xB

(119862119873minus1)1198711+120575119896(w)) ge 119871

119896+ 1

11990310158401015840= 120588 (yB

(119862119873minus1)1198711+120575119896(w)) ge 119871119896 + 1

(127)


1199031015840(x) and in B

11990310158401015840(y) are


1)1198711+120575

119896minus 2 ge 119877 minus 2119862

1198731198711+120575

119896










(119877minus21198621198731198711+120575119896 minus2119871119896)(119871119896+1)

sdot e119871120573

119896+1] = 119898(1 +1

2119871minus120591

119896)

119869119871119896

119871119896+ 1119877(1 minus

3119862119873

119877

sdot 119871119896) minus 119871

120573

119896+1= 119898119877[(1 +

1

2119871minus120591

119896)

119869


119896)


119873119871119896

1198711+984858

119896

) minus119871120573

119896+1

119898119877] = 119898119877[(1 +

1

2119871minus120591

119896)

119869


119896) (1 minus 3119862

119873119871minus984858

119896) minus

119871120573

119896+1

1198981198711+984858

119896

] ge 119898(1 +1

4


119896)

119869

119877 ge 120574 (119898 119871119896+1) 119877

(128)






1 119909

119873) 119909





1003816100381610038161003816 le 119862119889119871119889 (129)





by the functions

Φa =1

radic119873

sum

120587isinS119873

119873

⨂

119895=1

1119886120587minus1(119895) (130)

where a = 1198861 119886

119873 with 119886

1 119886

119873 = 119873



set is

ZNgt

= a = 1198861 119886

119873 119886

119895isinZ 119886

1 119886

119873 = 119873

(131)



1 119888

2 119888

119873 b = 119887

1 119888

2 119888

119873 with

1198861 119887

1 119888

2 119888

119873 = 119873 + 1

(ii) dZ(1198861 1198871) = 1


120588 (x y) = min120587isinS119873

max1le119895le119873

dZ (119909120587minus1(119895) 119910119895) (132)




119892V(x 120596) + U(x) where

U (x) = sum119894 =119895

119880(d (119909119894 119909

119895)) 119880 N 997888rarr R (133)


V (x 120596) = 119881 (1199091 120596) + sdot sdot sdot + 119881 (119909

119873 120596) (134)






Appendix



V (1199091 119909

119873 120596) = sum

119910isinxn119910119881 (119910 120596) (A1)





119908(y)) 119881 (119908 120596)

+ sum

V =119908

119888V119881 (V 120596) (A2)


119888V = nV (x) minus nV (y) (A3)



119908 while

n119908(x) minus n

119908(y) = 119888

119908isin Z 0 hence 1003816100381610038161003816119888119908

1003816100381610038161003816 ge 1 (A4)

Therefore

P 1003816100381610038161003816119892V (x 120596) minus 119892V (y 120596)

1003816100381610038161003816 le 119904

= E [P 1003816100381610038161003816V (x 120596) minus V (y 120596)

1003816100381610038161003816 le10038161003816100381610038161198921003816100381610038161003816

minus1119904 | F

=119908]

= E [P 1003816100381610038161003816119888119908119881 (119908 120596) | z (120596)

1003816100381610038161003816 le10038161003816100381610038161198921003816100381610038161003816

minus1119904 | F

=119908]

(A5)




120596 119888119908119881 (119908 120596) isin [z (120596) minus

10038161003816100381610038161198921003816100381610038161003816

minus1119904 z (120596) +

10038161003816100381610038161198921003816100381610038161003816

minus1119904]

= 120596 119881 (119908 120596)

isin [z (120596) minus11990410038161003816100381610038161198921003816100381610038161003816

minus1

119888119908

z (120596) +11990410038161003816100381610038161198921003816100381610038161003816

minus1

119888119908

]

(A6)


P 1003816100381610038161003816V (x 120596) minus V (y 120596)

1003816100381610038161003816 le10038161003816100381610038161198921003816100381610038161003816

minus1119904

le sup119886isinR

(119865119881119908(119886 + 2119904) minus 119865

119881119908(119886))

(A7)


at 119908 given F=119908 Since 119865




1003816100381610038161003816 le 119904 (A8)


1198710(u) admits



0 To prove the




100381610038161003816100381610038161003816GB1198710 (u)

(x y 119864)100381610038161003816100381610038161003816le eminus119898(119904)d(xy)

(A9)





119899

119895=11198612119871(119909

119895) If




1198711+120575(x) can be


(⋃

119895isinJ

1198612119871(119909

119895))∐(⋃

119894isinJ119888

1198612119871(119909

119894)) (A10)


120588 (119861119871(119909

119894) 119861

119871(119909

119895)) gt 2119871 (A11)


119873119871 then for some 119895

∘ 120588(119909

119895∘ 119910

119895∘) gt 119862

119873119871


120588 (119909119895 119909

119895∘) le diamΠx le 119860

119873119871

120588 (119910119895 119910

119895∘) le diamΠy lt 119860

119873119871

(A12)


120588 (119909119894 119910

119895) ge 120588 (119909

119895∘ 119910

119895∘) minus 120588 (119909

119894 119909

119895∘) minus 120588 (119910

119895 119910

119895∘)

ge 119862119873119871 minus 2119860

119873119871 ge 3119871

(A13)


yielding

120588 (Π119861119871(x) Π119861

119871(y)) gt 3119871 minus 2119871 = 119871 (A14)



Acknowledgments


References

































Volume 2014




Journal of











Journal of


Function Spaces






Algebra














119896also large)

1198711+984858

119896≫ 119871

1+120575


120588 (xB(119862119873minus1)119871119896

(w)) + 120588 (yB(119862119873minus1)119871119896(w))

ge 119877 minus (2119862119873minus 2) 119871

1+120575

119896

(126)

Assume first that

1199031015840= 120588 (xB

(119862119873minus1)1198711+120575119896(w)) ge 119871

119896+ 1

11990310158401015840= 120588 (yB

(119862119873minus1)1198711+120575119896(w)) ge 119871119896 + 1

(127)


1199031015840(x) and in B

11990310158401015840(y) are


1)1198711+120575

119896minus 2 ge 119877 minus 2119862

1198731198711+120575

119896










(119877minus21198621198731198711+120575119896 minus2119871119896)(119871119896+1)

sdot e119871120573

119896+1] = 119898(1 +1

2119871minus120591

119896)

119869119871119896

119871119896+ 1119877(1 minus

3119862119873

119877

sdot 119871119896) minus 119871

120573

119896+1= 119898119877[(1 +

1

2119871minus120591

119896)

119869


119896)


119873119871119896

1198711+984858

119896

) minus119871120573

119896+1

119898119877] = 119898119877[(1 +

1

2119871minus120591

119896)

119869


119896) (1 minus 3119862

119873119871minus984858

119896) minus

119871120573

119896+1

1198981198711+984858

119896

] ge 119898(1 +1

4


119896)

119869

119877 ge 120574 (119898 119871119896+1) 119877

(128)






1 119909

119873) 119909





1003816100381610038161003816 le 119862119889119871119889 (129)





by the functions

Φa =1

radic119873

sum

120587isinS119873

119873

⨂

119895=1

1119886120587minus1(119895) (130)

where a = 1198861 119886

119873 with 119886

1 119886

119873 = 119873



set is

ZNgt

= a = 1198861 119886

119873 119886

119895isinZ 119886

1 119886

119873 = 119873

(131)



1 119888

2 119888

119873 b = 119887

1 119888

2 119888

119873 with

1198861 119887

1 119888

2 119888

119873 = 119873 + 1

(ii) dZ(1198861 1198871) = 1


120588 (x y) = min120587isinS119873

max1le119895le119873

dZ (119909120587minus1(119895) 119910119895) (132)




119892V(x 120596) + U(x) where

U (x) = sum119894 =119895

119880(d (119909119894 119909

119895)) 119880 N 997888rarr R (133)


V (x 120596) = 119881 (1199091 120596) + sdot sdot sdot + 119881 (119909

119873 120596) (134)






Appendix



V (1199091 119909

119873 120596) = sum

119910isinxn119910119881 (119910 120596) (A1)





119908(y)) 119881 (119908 120596)

+ sum

V =119908

119888V119881 (V 120596) (A2)


119888V = nV (x) minus nV (y) (A3)



119908 while

n119908(x) minus n

119908(y) = 119888

119908isin Z 0 hence 1003816100381610038161003816119888119908

1003816100381610038161003816 ge 1 (A4)

Therefore

P 1003816100381610038161003816119892V (x 120596) minus 119892V (y 120596)

1003816100381610038161003816 le 119904

= E [P 1003816100381610038161003816V (x 120596) minus V (y 120596)

1003816100381610038161003816 le10038161003816100381610038161198921003816100381610038161003816

minus1119904 | F

=119908]

= E [P 1003816100381610038161003816119888119908119881 (119908 120596) | z (120596)

1003816100381610038161003816 le10038161003816100381610038161198921003816100381610038161003816

minus1119904 | F

=119908]

(A5)




120596 119888119908119881 (119908 120596) isin [z (120596) minus

10038161003816100381610038161198921003816100381610038161003816

minus1119904 z (120596) +

10038161003816100381610038161198921003816100381610038161003816

minus1119904]

= 120596 119881 (119908 120596)

isin [z (120596) minus11990410038161003816100381610038161198921003816100381610038161003816

minus1

119888119908

z (120596) +11990410038161003816100381610038161198921003816100381610038161003816

minus1

119888119908

]

(A6)


P 1003816100381610038161003816V (x 120596) minus V (y 120596)

1003816100381610038161003816 le10038161003816100381610038161198921003816100381610038161003816

minus1119904

le sup119886isinR

(119865119881119908(119886 + 2119904) minus 119865

119881119908(119886))

(A7)


at 119908 given F=119908 Since 119865




1003816100381610038161003816 le 119904 (A8)


1198710(u) admits



0 To prove the




100381610038161003816100381610038161003816GB1198710 (u)

(x y 119864)100381610038161003816100381610038161003816le eminus119898(119904)d(xy)

(A9)





119899

119895=11198612119871(119909

119895) If




1198711+120575(x) can be


(⋃

119895isinJ

1198612119871(119909

119895))∐(⋃

119894isinJ119888

1198612119871(119909

119894)) (A10)


120588 (119861119871(119909

119894) 119861

119871(119909

119895)) gt 2119871 (A11)


119873119871 then for some 119895

∘ 120588(119909

119895∘ 119910

119895∘) gt 119862

119873119871


120588 (119909119895 119909

119895∘) le diamΠx le 119860

119873119871

120588 (119910119895 119910

119895∘) le diamΠy lt 119860

119873119871

(A12)


120588 (119909119894 119910

119895) ge 120588 (119909

119895∘ 119910

119895∘) minus 120588 (119909

119894 119909

119895∘) minus 120588 (119910

119895 119910

119895∘)

ge 119862119873119871 minus 2119860

119873119871 ge 3119871

(A13)


yielding

120588 (Π119861119871(x) Π119861

119871(y)) gt 3119871 minus 2119871 = 119871 (A14)



Acknowledgments


References

































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












Appendix



V (1199091 119909

119873 120596) = sum

119910isinxn119910119881 (119910 120596) (A1)





119908(y)) 119881 (119908 120596)

+ sum

V =119908

119888V119881 (V 120596) (A2)


119888V = nV (x) minus nV (y) (A3)



119908 while

n119908(x) minus n

119908(y) = 119888

119908isin Z 0 hence 1003816100381610038161003816119888119908

1003816100381610038161003816 ge 1 (A4)

Therefore

P 1003816100381610038161003816119892V (x 120596) minus 119892V (y 120596)

1003816100381610038161003816 le 119904

= E [P 1003816100381610038161003816V (x 120596) minus V (y 120596)

1003816100381610038161003816 le10038161003816100381610038161198921003816100381610038161003816

minus1119904 | F

=119908]

= E [P 1003816100381610038161003816119888119908119881 (119908 120596) | z (120596)

1003816100381610038161003816 le10038161003816100381610038161198921003816100381610038161003816

minus1119904 | F

=119908]

(A5)




120596 119888119908119881 (119908 120596) isin [z (120596) minus

10038161003816100381610038161198921003816100381610038161003816

minus1119904 z (120596) +

10038161003816100381610038161198921003816100381610038161003816

minus1119904]

= 120596 119881 (119908 120596)

isin [z (120596) minus11990410038161003816100381610038161198921003816100381610038161003816

minus1

119888119908

z (120596) +11990410038161003816100381610038161198921003816100381610038161003816

minus1

119888119908

]

(A6)


P 1003816100381610038161003816V (x 120596) minus V (y 120596)

1003816100381610038161003816 le10038161003816100381610038161198921003816100381610038161003816

minus1119904

le sup119886isinR

(119865119881119908(119886 + 2119904) minus 119865

119881119908(119886))

(A7)


at 119908 given F=119908 Since 119865




1003816100381610038161003816 le 119904 (A8)


1198710(u) admits



0 To prove the




100381610038161003816100381610038161003816GB1198710 (u)

(x y 119864)100381610038161003816100381610038161003816le eminus119898(119904)d(xy)

(A9)





119899

119895=11198612119871(119909

119895) If




1198711+120575(x) can be


(⋃

119895isinJ

1198612119871(119909

119895))∐(⋃

119894isinJ119888

1198612119871(119909

119894)) (A10)


120588 (119861119871(119909

119894) 119861

119871(119909

119895)) gt 2119871 (A11)


119873119871 then for some 119895

∘ 120588(119909

119895∘ 119910

119895∘) gt 119862

119873119871


120588 (119909119895 119909

119895∘) le diamΠx le 119860

119873119871

120588 (119910119895 119910

119895∘) le diamΠy lt 119860

119873119871

(A12)


120588 (119909119894 119910

119895) ge 120588 (119909

119895∘ 119910

119895∘) minus 120588 (119909

119894 119909

119895∘) minus 120588 (119910

119895 119910

119895∘)

ge 119862119873119871 minus 2119860

119873119871 ge 3119871

(A13)


yielding

120588 (Π119861119871(x) Π119861

119871(y)) gt 3119871 minus 2119871 = 119871 (A14)



Acknowledgments


References

































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










yielding

120588 (Π119861119871(x) Π119861

119871(y)) gt 3119871 minus 2119871 = 119871 (A14)



Acknowledgments


References

































Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









Documents

Research Article Direct Scaling Analysis of Fermionic Multiparticle … · 2019. 7. 30. · obtained for a xed energy (cf. [, ]) or in an energy interval (cf. [, ]) into the language