Research ArticleSobolev Spaces on Locally Compact Abelian GroupsCompact Embeddings and Local Spaces

PrzemysBaw Goacuterka1 Tomasz Kostrzewa1 and Enrique G Reyes2

1 Department of Mathematics and Information Sciences Warsaw University of Technology Ul Koszykowa 75 00-662Warsaw Poland2Departamento de Matematica y Ciencia de la Computacion Universidad de Santiago de Chile Casilla 307 Correo 2 Santiago Chile

Correspondence should be addressed to Przemysław Gorka pgorkaminipwedupl

Received 23 May 2013 Accepted 20 December 2013 Published 9 February 2014

Academic Editor Kehe Zhu

Copyright copy 2014 Przemysław Gorka et al This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

We continue our research on Sobolev spaces on locally compact abelian (LCA) groups motivated by our work on equations withinfinitely many derivatives of interest for string theory and cosmology In this paper we focus on compact embedding resultsand we prove an analog for LCA groups of the classical Rellich lemma and of the Rellich-Kondrachov compactness theoremFurthermore we introduce Sobolev spaces on subsets of LCA groups and study its main properties including the existence ofcompact embeddings into 119871119901-spaces

1 Introduction

In this paper we continue our research on Sobolev spaceson locally compact abelian groups [1 2] and we examineanalogs of the Rellich lemma and the Rellich-Kondrachovcompactness theorem Sobolev spaces are well understood ondomains of R119899 see [3 4] compact Riemannian manifolds[5 6] and metric measure spaces [7ndash9] There are also someworks on Sobolev spaces in the 119901-adic context see [10 11]and references therein and in special cases of locally compactgroups such as the Heisenberg group [12]

We are interested in Sobolev spaces in this general contextdue to our work on nonlinear equations ldquoin infinitely manyderivativesrdquo of interest for contemporary physical theories in[13ndash15] two of the present authors in collaboration with HPrado have investigated the existence of regular solutions tothe generalized Euclidean Bosonic string equation

Δ119890minus119888Δ

120601 = 119880 (119909 120601) 119888 gt 0 (1)

and some of its generalizations and in [16 17] the sameresearchers have developed a functional calculus appropriatefor the study of the initial value problem for ldquoordinaryrdquoequations of the form

119891 (120597119905) 120601 = 119869 (119905) (2)

Equations such as (1) and (2) are specially interesting forstring theory and cosmology see [18ndash21] and referencestherein These two areas are undergoing such a fast devel-opment that it seems important to understand (1) and (2)in contexts beyond the usual geometric arena of analysis on(Riemannian) manifolds We think that topological groupsare a natural testing ground for gathering a better under-standing of (1) and (2) For instance this setting would allowus to consider (1) for functions 120601 on finite spaces with groupstructure (see eg [22]) or for functions depending on aninfinite number of independent variables On the other handthis generalization makes it necessary to develop a theoryof Sobolev spaces on LCA groups appropriate for the studyof nonlocal equations along the lines of [13ndash15] It is indeedpossible to do so essentially because of the existence of groupstructure and the availability of Fourier transform

We introduced Sobolev spaces on LCA groups in [1] Inthat reference we proved analogs of the Sobolev embeddingand Rellich-Kondrachov theorems and we used these resultsto prove the existence of regular solutions to (1) on compactabelian groups Then in [2] we considered a version of theclassical Rellich lemma and presented another theorem onregular solutions to (1) Now our version of the Rellichlemma appearing in [2] relies on a technical assumptionon the structure of the group of characters of the given

Hindawi Publishing CorporationJournal of Function SpacesVolume 2014 Article ID 404738 6 pageshttpdxdoiorg1011552014404738

2 Journal of Function Spaces

group 119866 which limits its applicability In this paper weremove this assumption and prove a version of the Rellichlemma which can be applied in great generality and we alsoimprove our original Rellich-Kondrachov theorem provenin [1] Moreover we introduce Sobolev spaces on subsets ofLCA groups in analogy with the Sobolev spaces defined ondomains ofR119899 As in this classical case we expect these spacesto be useful in the study of differential equations and otherapplications [23]

We organize this paper as follows In Section 2 we recallour definition of Sobolev spaces and our previous embeddingand compactness results In Section 3 we state and proveour new compactness results and in Section 4 we discussSobolev spaces on subsets of LCA groups

We use standard notations from harmonic analysis [2425] Let us fix a locally compact abelian group 119866 We denoteby 119889119909 the uniqueHaarmeasure of119866 and by119866and the dual groupof the group119866 that is119866and is the locally compact abelian groupof all continuous group homomorphisms from119866 to the circlegroup 119879 The 119871119901 spaces over 119866 are defined as usual

119871119901(119866) = 119891 119866 997888rarr C int

119866

1003816100381610038161003816119891 (119909)1003816100381610038161003816

119901119889119909 lt infin (3)

and we set

10038171003817100381710038171198911003817100381710038171003817119871119901(119866)

= (int119866

1003816100381610038161003816119891 (119909)1003816100381610038161003816

119901119889119909)

1119901

(4)

We also recall that the Fourier transform on 119866 is definedas follows if 119891 isin 119871

1(119866) then its Fourier transform is the

function 119891 119866andrarr C given by

119891 (120585) = int119866

120585(119909)119891 (119909) 119889119909 (5)

We consider general LCA groups in Section 2 but werestrict ourselves to compact abelian groups when provingcompactness results in Section 3

2 Sobolev Spaces

We introduce Sobolev spaces following our previous papers[1 2]Our definition uses essentially the Fourier transform forLCA groups and as explained in [1] it generalizes naturallythe standard notions of Sobolev spaces in the case of T119899 andR119899 see [26] and [4 Chapter 4]

We denote by Γ the set

Γ = 120574 119866and997888rarr [0infin) exist119888120574

forall120572120573isin119866and120574 (120572120573)

le 119888120574 [120574 (120572) + 120574 (120573)]

(6)

Definition 1 Let us fix a map 120574 isin Γ and a nonnegative realnumber 119904 We will say that 119891 isin 119871

2(119866) belongs to the Sobolev

space119867119904

120574(119866) if the following integral is finite

int119866and

(1 + 120574(120585)2)11990410038161003816100381610038161003816119891 (120585)

10038161003816100381610038161003816

2

119889120585 (7)

Moreover for 119891 isin 119867119904

120574(119866) its norm 119891

119867119904120574(119866)

is defined asfollows

10038171003817100381710038171198911003817100381710038171003817119867119904120574(119866)

= (int119866and

(1 + 120574(120585)2)11990410038161003816100381610038161003816119891 (120585)

10038161003816100381610038161003816

2

119889120585)

12

(8)

Remark 2 Aparticular instance ofDefinition 1 appears in thepaper [26] by Feichtinger and Werther Another particularcase of our definition is in [27] We also note that in 119901-adicanalysis Sobolev spaces are defined in a way analogous to ourDefinition 1 if we take 120574(120585) = 120585119901 where sdot 119901 is a 119901-adicnorm on Q119899

119901≃ Q119899and

119901 then (7) and (8) allow us to recover the

119901-adic Sobolev spaces considered in [11]

Remark 3 The fact that 120574 isin Γ implies that our spaces119867119904

120574(119866)

are Banach algebras under some assumptions on 119904 see ourprevious paper [1]

We recall the following results proven in [1]

Proposition 4 Let 119866 be a locally compact abelian group Onehas the following

(1) Consider 119867119904

120574(119866) 997893rarr 119871

2(119866) Moreover for each 119891 isin

119867119904

120574(119866) the following inequality holds

100381710038171003817100381711989110038171003817100381710038171198712(119866)

le1003817100381710038171003817119891

1003817100381710038171003817119867119904120574(119866)

(9)

(2) If 119904 gt 120590 then 119867119904

120574(119866) 997893rarr 119867

120590

120574(119866) Moreover the

following inequality holds1003817100381710038171003817119891

1003817100381710038171003817119867120590120574(119866)

le1003817100381710038171003817119891

1003817100381710038171003817119867119904120574(119866)

(10)

Theorem 5 Let119866 be a locally compact abelian group One hasthe following

(1) If 1(1 + 120574(sdot)2) isin 119871

119904(119866

and) then

119867119904

120574(119866) 997893rarr 119862 (119866) (11)

inwhich119862(119866) denotes the space of continuous complex-valued functions on119866Moreover there exists a constant119862(120574 119904) such that for each 119891 isin 119867

119904

120574(119866) the following

inequality holds1003817100381710038171003817119891

1003817100381710038171003817119862(119866)le 119862 (120574 119904)

10038171003817100381710038171198911003817100381710038171003817119867119904120574(119866)

(12)

(2) If 120572 gt 119904 and 1(1 + 120574(sdot)2) isin 119871

120572(119866

and) then

119867119904

120574(119866) 997893rarr 119871

120572lowast

(119866) (13)

where120572lowast= 2120572(120572minus119904)Moreover there exists a constant

119863(120574 119904) such that for each 119891 isin 119867119904

120574(119866) the following

inequality holds1003817100381710038171003817119891

1003817100381710038171003817119871120572lowast(119866)

le 119863 (120574 119904)1003817100381710038171003817119891

1003817100381710038171003817119867119904120574(119866)

(14)

Theorem 5 is our version of the classical Sobolev embed-ding theorem appearing for instance in [3] for the case 119866 =

R119899

Journal of Function Spaces 3

3 Compact Embedding

We recall that the notation 119860 997893rarr997893rarr 119861 means that the space119860 is compactly embedded in 119861 We begin with our refinedversion of the classical Rellich lemma

Theorem 6 Let 119866 be a compact group If lim120585rarrinfin120574(120585) = infinthat is for each119872 gt 0 there exists finite set119880 such that for any120585 isin 119866

and 119880 one has 120574(120585) ge 119872 then for all 119904 gt 120575

119867119904

120574(119866) 997893rarr997893rarr 119867

120575

120574(119866) (15)

Proof We begin the proof stating a classical fact (see eg[28]) on the characterization of precompact sets in 119871

119901(119867)

spaces

Theorem7 (see [28]) LetΦ be a family of functions in 119871119901(119867)

1 le 119901 lt infin Then Φ is compact in 119871119901(119867) if and only if the

following conditions hold

(i) There exists 119872 gt 0 such that 120601119871119901(119867)

le 119872 for all120601 isin Φ

(ii) For every 120576 gt 0 there exists compact set 119870 in 119866 suchthat for each 120601 isin Φ

10038171003817100381710038171206011003817100381710038171003817119871119901(119867119870)

le 120576 (16)

(iii) For all 120576 gt 0 there exists unit neighborhood 119881 suchthat for all 119909 isin 119881 and all 120601 isin Φ

1003817100381710038171003817119871119909120601 minus 1206011003817100381710038171003817119871119901(119867)

le 120576 (17)

Now we return to the proof of Theorem 6 Let 119891119899 be abounded sequence in 119867

119904

120574(119866) 119891119899119867119904

120574(119866)

le 119872 We need toshow that there exists subsequence that converges strongly in119867

120575

120574(119866) We will prove this fact by showing that the following

sequence

119892119899 (120585) =10038161003816100381610038161003816119891119899 (120585)

10038161003816100381610038161003816(1 + 120574

2(120585))

1205752 (18)

is compact in 1198712(119866

and) We use Weilrsquos theorem since 120575 lt 119904 we

get

1003817100381710038171003817119892119899

10038171003817100381710038171198712(119866and)le1003817100381710038171003817119891119899

1003817100381710038171003817119867119904120574(119866)

le 119872 (19)

and so (i) holds Now we consider condition (ii) Let us fix120598 gt 0 We consider two cases if 119866and is finite group then wecan take simply 119870 = 119866

and and condition (ii) is satisfied Onthe other hand if the dual group 119866

and is infinite it is enoughto recall that if 119866 is a compact group then by the Pontryaginduality theorem its dual 119866and is discrete and therefore everycompact set must be finite From our assumption we can finda compact set 119870 such that

1

(1 + 1205742 (120585))119904minus120575

le120598

1198722 for all 120585 isin 119866

and 119870 (20)

Hence

int119866and119870

10038161003816100381610038161003816119891119899 (120585)

10038161003816100381610038161003816

2

(1 + 1205742(120585))

120575

119889120585

= int119866and119870

10038161003816100381610038161003816119891119899 (120585)

10038161003816100381610038161003816

2 (1 + 1205742(120585))

119904

(1 + 1205742 (120585))119904minus120575

119889120585

le sup120585isin119866and119870

1

(1 + 1205742 (120585))119904minus120575

int119866and119870

10038161003816100381610038161003816119891119899 (120585)

10038161003816100381610038161003816

2

(1 + 1205742(120585))

119904

119889120585

le1003817100381710038171003817119891119899

1003817100381710038171003817

2

119867119904120574(119866)

sup120585isin119866and119870

1

(1 + 1205742 (120585))119904minus120575

le 120598

(21)

and so (ii) holds It remains to check condition (iii) Since 119866and

is discrete and each set is open we can take 119881 = 119890 where 119890is unit in 119866

and Thus condition (iii) is satisfied and Theorem 6follows fromWeilrsquos result

Theorem 6 appears in our previous paper [2] under theadditional assumption that the dual group 119866and is countable

Now we consider embeddings of 119867119904

120574(119866) into 119862(119866) and

119871119901(119866)We proved in [1] that119867119904

120574(119866) is continuously embedded

in 119862(119866) We prove in Theorem 10 below that if 119866 is compactthen 119867

119904

120574997893rarr997893rarr 119862(119866) and finally in Theorem 11 we consider

a version of the Rellich-Kondrachov which refines an analo-gous result from [1] We need the following lemma

Lemma 8 Let 119867 be a discrete group and 119891 isin 1198711(119867) Then

for every 120576 gt 0 there exists a finite set 119880 such that for anyℎ isin 119867 119880 one has |119891(ℎ)| le 120576

Proof Let us suppose that the theorem is not true so thatthere is a number 120576 gt 0 such that for every finite set 119880there exists ℎ isin 119867 119880 with |119891(ℎ)| ge 120576 Let 1198801 be a finiteset and let ℎ1 isin 119867 1198801 be such that |119891(ℎ1)| ge 120576 We define1198802 = 1198801cupℎ1 Since1198802 is finite there exists ℎ2 isin 1198671198802 suchthat |119891(ℎ2)| ge 120576 By induction we get a sequence of sets 119880119894

and ℎ119894 for 119894 = 1 2 such that119880119894+1 = 119880119894 cup ℎ119894 and for each119894 we have |119891(ℎ119894)| ge 120576 Since Haar measure on discrete groupsis a multiple of the counting measure we get

int119867

1003816100381610038161003816119891 (ℎ)1003816100381610038161003816 119889ℎ = int

119867119880

1003816100381610038161003816119891 (ℎ)1003816100381610038161003816 119889ℎ + int

119880

1003816100381610038161003816119891 (ℎ)1003816100381610038161003816 119889ℎ

ge

infin

sum

119894=1

1003816100381610038161003816119891 (ℎ119894)1003816100381610038161003816 ge

infin

sum

119894=1

120576 = +infin

(22)

where 119880 = ⋃infin

119894=1119880119894 and this is a contradiction

Remark 9 We note that if 119867 is countable the proof ofLemma 8 is elementary int

119867|119891(ℎ)|119889ℎ = 119886 sum

infin

119894=1|119891(ℎ119894)| for

some 119886 isin C and suminfin

119894=1|119891(ℎ119894)| lt infin implies that given 120598 gt 0

there exists 119870 gt 0 such that suminfin

119894=119870|119891(ℎ119894)| lt 120598 The result then

follows

Theorem10 If119866 is compact 1(1+1205742(sdot)) isin 119871120572(119866

and) and 119904 lt 120572

then119867

119904

120574(119866) 997893rarr997893rarr 119862 (119866) (23)

4 Journal of Function Spaces

Proof Using Lemma 8 for the function 1(1 + 1205742(sdot))

120572 we seethat 120574 satisfies the assumptions ofTheorem 6 so for 119904 lt 120572 weget that

119867119904

120574(119866) 997893rarr997893rarr 119867

120572

120574(119866) (24)

Moreover thanks to the first part of Theorem 5 we have

119867120572

120574(119866) 997893rarr 119862 (119866) (25)

and the proof follows

Theorem11 If119866 is compact 1(1+1205742(sdot)) isin 119871120572(119866

and) and 119904 lt 120572

then

119867119904

120574(119866) 997893rarr997893rarr 119871

119901(119866) (26)

for all 119901 lt 120572lowast= 2120572(120572 minus 119904)

Proof Let us take 120575 = 120572 minus 2120572119901 then 120575 lt 119904 Thus byTheorem 6 we have the compact embedding

119867119904

120574(119866) 997893rarr997893rarr 119867

120575

120574(119866) (27)

Next using the second part of Theorem 5 we have thecontinuous embedding

119867120575

120574(119866) 997893rarr 119871

120572lowast

120575 (119866) (28)

where 120572lowast

120575= 2120572(120572 minus 120575) = 119901

4 Sobolev Spaces on Subsets of LCA Groups

In this section we deal with Sobolev spaces defined onsubsets of locally compact abelian groups As mentioned inSection 1 we are motivated by analogous studies of functionspaces on domains of R119899 (see eg [29]) by the fact thatinteresting applications exist [23] and by the possibility ofusing them as tools for the study of differential equations onsubsets of LCA groupsWe start with the following definition

Definition 12 Let119880 be a subset of a LCA group 119866 We definethe Sobolev space 119867

119904

120574(119880) sub 119871

2(119880) as the space of all 119891 isin

1198712(119880) such that there exists 119865 isin 119867

119904

120574(119866) with 119865|119880 = 119891 and

we equip it with the norm

10038171003817100381710038171198911003817100381710038171003817119867119904120574(119880)

= inf 119865119867119904120574(119866) 119865 isin 119867

119904

120574(119866) and 119865|119880 = 119891

(29)

An analogous definition (of spaces119861119904

119901119902on domains ofR119899)

appears in [29] see his Definition 23 It can be easily shownthat 119867119904

120574(119880) is a Banach space We will say that it is a local

Sobolev spaceUsing appropriate embeddings for119867119904

120574(119866) and the defini-

tion of119867119904

120574(119880) we can prove the following

Theorem 13 Let 119866 be a locally compact abelian group and let119880 sub 119866 Then we have

(1) The continuous inclusion 119867119904

120574(119880) 997893rarr 119871

2(119880) holds

Moreover for each 119891 isin 119867119904

120574(119880) the following inequality

holds1003817100381710038171003817119891

10038171003817100381710038171198712(119880)le1003817100381710038171003817119891

1003817100381710038171003817119867119904120574(119880)

(30)

(2) If 119904 gt 120590 then119867119904

120574(119866) 997893rarr 119867

120590

120574(119880) Moreover the follow-

ing inequality holds1003817100381710038171003817119891

1003817100381710038171003817119867120590120574(119880)

le1003817100381710038171003817119891

1003817100381710038171003817119867119904120574(119880)

(31)

(3) If 1(1 + 120574(sdot)2) isin 119871

119904(119866

and) then 119867

119904

120574(119880) 997893rarr 119862(119880)

Moreover there exists a constant 119862(120574 119904) such that foreach 119891 isin 119867

119904

120574(119880) the following inequality holds

10038171003817100381710038171198911003817100381710038171003817119862(119880)

le 119862 (120574 119904)1003817100381710038171003817119891

1003817100381710038171003817119867119904120574(119880)

(32)

(4) If 120572 gt 119904 and 1(1 + 120574(sdot)2) isin 119871

120572(119866

and) then 119867

119904

120574(119880) 997893rarr

119871120572lowast

(119880) where 120572lowast= 2120572(120572 minus 119904) Moreover there exists

a constant 119863(120574 119904) such that for each 119891 isin 119867119904

120574(119866) the

following inequality holds1003817100381710038171003817119891

1003817100381710038171003817119871120572lowast(119880)

le 119863 (120574 119904)1003817100381710038171003817119891

1003817100381710038171003817119867119904120574(119880)

(33)

We now prove the following compactness theorem indetail

Theorem 14 Let119866 be an LCA group and let119880 be a subset of119866of finite measure Assume that 1(1 + 1205742(sdot)) isin 119871

120572(119866

and) for some

120572 gt 119904 and that1003816100381610038161003816120585 (ℎ) minus 1

1003816100381610038161003816

(1 + 1205742 (120585))119904 997888rarr

ℎrarr1198900 119906119899119894119891119900119903119898119897119910 119908119894119905ℎ 119903119890119904119901119890119888119905 119905119900 120585 isin 119866

and

(34)

Then for all 119901 lt 120572lowast one has the compact embedding

119867119904

120574(119880) 997893rarr997893rarr 119871

119901(119880) (35)

The convergence concept used in (34) is explained in ourprevious paper [1] Let us mention that a condition similarto (34) appears in the characterization of precompact sets in1198712(119866) via Fourier transform see [30]

Proof We will need two lemmas which we proved in [1]

Lemma 15 Let 119865 isin 119867119904

120574(119866) and assume that

1003816100381610038161003816120585 (ℎ) minus 11003816100381610038161003816

(1 + 1205742 (120585))119904 997888rarr

ℎrarr1198900 119906119899119894119891119900119903119898119897119910 119908119894119905ℎ 119903119890119904119901119890119888119905 119905119900 120585 isin 119866

and

(36)

Then for every ℎ isin 119866

int119866

|119865 (119909ℎ) minus 119865 (119909)|2119889119909 le 119862 (ℎ) 119865

2

119867119904120574(119866)

(37)

where 119862(ℎ) rarrℎrarr119890

0

Journal of Function Spaces 5

Lemma 16 Let us denote by I the set of all symmetric unitneighbourhoods of group G partially ordered by inclusion Let(120601119881)119881isinI be a Dirac net and let 119865 isin 119867

119904

120574(119866) Then

int119866

1003816100381610038161003816119865 lowast 120601119881 (119909) minus 119865 (119909)1003816100381610038161003816

2119889119909 le 119865119867119904

120574(119866)sup

119910isin119881

119862 (119910) (38)

Nowwe can continuewith the proof ofTheorem 14 Let119891119899

be a bounded sequence in119867119904

120574(119880) that is 119891119899119867119904

120574(119880)

le 119862 Fromthe very definition there exist functions 119865119899 such that 119865119899 = 119891119899

in119880119865119899 isin 119867119904

120574(119866) and 119865119899119867119904

120574(119866)

le 119891119899119867119904120574(119880)

+1 for 119899 = 1 2 Now by Theorem 13 we get that the sequence 119891119899 is boundedin 119871

120572lowast

(119880) Hence there exists a subsequence 119891119899119896such that

119891119899119896 119891 in 119871

120572lowast

(119880) and there also exists a subsequence 119865119899119896

such that 119865119899119896 119865 in 119871

120572lowast

(119866) For simplicity we simply write119891119899 and 119865119899 for these subsequences From the basic propertiesof weak convergence we have that 119865 = 119891 in 119880 We will showthat for 119902 lt 120572

lowast we have

119891119899 997888rarr 119891 in 119871119902(119880) (39)

Let us set 119865(119881) = 119865 lowast 120601119881 From the triangle inequality we get

1003817100381710038171003817119891119899 minus 11989110038171003817100381710038171198712(119880)

le1003817100381710038171003817119891119899 minus 119865119899(119881)

10038171003817100381710038171198712(119880)

+1003817100381710038171003817119865119899(119881) minus 119865(119881)

10038171003817100381710038171198712(119880)+1003817100381710038171003817119865(119881) minus 119891

10038171003817100381710038171198712(119880)

le1003817100381710038171003817119865119899 minus 119865119899(119881)

10038171003817100381710038171198712(119866)+1003817100381710038171003817119865119899(119881) minus 119865(119881)

10038171003817100381710038171198712(119880)

+1003817100381710038171003817119865(119881) minus 119865

10038171003817100381710038171198712(119866)

(40)

From Lemma 15 we get

sup119899

int119866

10038161003816100381610038161003816119865119899(119881)

(119909) minus 119865119899 (119909)10038161003816100381610038161003816

2

119889119909 le sup119899

1003817100381710038171003817119865119899

1003817100381710038171003817

2

119867119904120574(119866)

sup119910isin119881

119862 (119910)

le (119862 + 1)2sup119910isin119881

119862 (119910)

(41)

Furthermore it can be easily shown that for each 120598 gt 0 thereexists 119881120598 such that for all 119881 isin I with 119881 sub 119881120598 we have

1003817100381710038171003817119865(119881) minus 11986510038171003817100381710038171198712(119866)

le120598

3 (42)

Hance we can choose 119881120598 such that

1003817100381710038171003817119891119899 minus 11989110038171003817100381710038171198712(119880)

le2

3120598 +

10038171003817100381710038171003817119865119899(119881120598)

minus 119865(119881120598)

100381710038171003817100381710038171198712(119880) (43)

We need to estimate 119865119899(119881120598)minus 119865(119881120598)

1198712(119880)

Since 119865119899 119865 in119871120572lowast

(119866) we get that 119865119899(119881120598)(119909) rarr

119899rarrinfin119865(119881120598)

(119909) Moreover wehave

10038161003816100381610038161003816119865119899(119881120598)

(119909) minus 119865(119881120598)(119909)

10038161003816100381610038161003816

2

=

10038161003816100381610038161003816100381610038161003816

int119866

(119865119899(119910) minus 119865(119910))120601119881120598(119910

minus1119909)119889119910

10038161003816100381610038161003816100381610038161003816

2

=

10038161003816100381610038161003816100381610038161003816

int119866

(119865119899(119910) minus 119865(119910))radic120601119881120598(119910minus1119909)radic120601119881120598

(119910minus1119909)119889119910

10038161003816100381610038161003816100381610038161003816

2

le int119866

120601119881120598(119910

minus1119909) 119889120583119866 (119910)

times int119866

1003816100381610038161003816119865119899 (119910) minus 119865 (119910)1003816100381610038161003816

2120601119881120598

(119910minus1119909) 119889119910

le sup119911isin119881120598

120601119881120598(119911) int

119866

1003816100381610038161003816119865119899 (119910) minus 119865 (119910)1003816100381610038161003816

2120601119881120598

(119910minus1119909) 119889119910

(44)

Hence since 119880 has a finite measure we get from Lebesgueconvergence theorem that

10038171003817100381710038171003817119865119899(119881120598)

minus 119865(119881120598)

100381710038171003817100381710038171198712(119880)le120598

3 (45)

We conclude that 119891119899 rarr 119891 in 1198712(119880) In order to finish the

proof it is enough to use the Vitali convergence theoremWeconclude that 119891119899 rarr 119891 in 119871

119902(119880) for 119902 lt 120572

lowast

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

Przemysław Gorkarsquos research is partially supported by theEuropean Union in the framework of European Social Fundthrough the Warsaw University of Technology DevelopmentProgram EnriqueG Reyesrsquo research is partially supported bythe Project FONDECYT no 1111042

References

[1] P Gorka and E G Reyes ldquoSobolev spaceson locally compactabelian groupsrdquo httparXivorgabs12083053 in press

[2] P Gorka T Kostrzewa and E G Reyes ldquoThe Rellich lemmaon compact abelian groups and equations of infinite orderrdquoInternational Journal of Geometric Methods in Modern Physicsvol 10 no 2 Article ID 1220030 11 pages 2013

[3] R A Adams Sobolev Spaces Academic Press New York NYUSA 1975

[4] M E Taylor Partial Differential Equations vol 23 SpringerNew York NY USA 1996

[5] T Aubin Some Nonlinear Problems in Riemannian GeometrySpringer Berlin Germany 1998

[6] E Hebey Sobolev Spaces on Riemannian Manifolds vol 1635Springer Berlin Germany 1996

[7] P Hajłasz ldquoSobolev spaces on an arbitrary metric spacerdquoPotential Analysis vol 5 no 4 pp 403ndash415 1996

[8] P Hajłasz and P Koskela ldquoSobolev met Poincarerdquo Memoirs ofthe American Mathematical Society vol 145 no 688 2000

[9] N Shanmugalingam ldquoNewtonian spaces an extension ofSobolev spaces to metric measure spacesrdquo Revista MatematicaIberoamericana vol 16 no 2 pp 243ndash279 2000

6 Journal of Function Spaces

[10] Y-C Kim ldquoA simple proof of the 119901-adic version of the Sobolevembedding theoremrdquo Korean Mathematical Society vol 25 no1 pp 27ndash36 2010

[11] J J Rodrıguez-Vega and W A Zuniga-Galindo ldquoElliptic pseu-dodifferential equations and Sobolev spaces over 119901-adic fieldsrdquoPacific Journal ofMathematics vol 246 no 2 pp 407ndash420 2010

[12] H Bahouri C Fermanian-Kammerer and I Gallagher ldquoAnal-yse de lrsquoespace des phases et calcul pseudo-differential sur legroupe de Heisenbergrdquo Comptes Rendus Mathematique vol347 no 17-18 pp 1021ndash1024 2009

[13] P Gorka H Prado and E G Reyes ldquoNonlinear equations withinfinitely many derivativesrdquo Complex Analysis and OperatorTheory vol 5 no 1 pp 313ndash323 2011

[14] P Gorka H Prado and E G Reyes ldquoGeneralized euclideanbosonic string equationsrdquo in Operator Theory Advances andApplications vol 224 pp 147ndash169 Springer Basel Switzerland2012

[15] P Gorka H Prado and E G Reyes ldquoOn a general class ofnonlocal equationsrdquo Annales Henri Poincare vol 14 no 4 pp947ndash966 2013

[16] P Gorka H Prado and E G Reyes ldquoFunctional calculusvia Laplace transform and equations with infinitely manyderivativesrdquo Journal of Mathematical Physics vol 51 no 10Article ID 103512 2010

[17] P Gorka H Prado and E G Reyes ldquoThe initial value problemfor ordinary differential equations with infinitely many deriva-tivesrdquo Classical and Quantum Gravity vol 29 no 6 Article ID065017 2012

[18] N Barnaby and N Kamran ldquoDynamics with infinitely manyderivatives the initial value problemrdquo Journal of High EnergyPhysics no 2 40 pages 2008

[19] G Calcagni M Montobbio and G Nardelli ldquoLocalization ofnonlocal theoriesrdquo Physics Letters B vol 662 no 3 pp 285ndash2892008

[20] V S Vladimirov ldquoOn the equation of a 119901-adic open string for ascalar tachyon fieldrdquo Izvestiya vol 69 no 3 pp 487ndash512 2005

[21] E Witten ldquoNoncommutative geometry and string field theoryrdquoNuclear Physics B vol 268 no 2 pp 253ndash294 1986

[22] D Zhou and C J C Burges ldquoHigh-order regularization ongraphsrdquo in Proceedings of the 6th International Workshop onMining and Learning with Graphs Helsinki Finland 2008httpwwwcishutfiMLG08paperspostersessionpaper39pdf

[23] H Triebel ldquoWavelets in function spaces on Lipschitz domainsrdquoMathematische Nachrichten vol 280 no 9-10 pp 1205ndash12182007

[24] E Hewitt andK A RossAbstract Harmonic Analysis Vol I DieGrundlehren der mathematischen Wissenschaften SpringerBerlin Germany 1963

[25] E Hewitt and K A Ross Abstract Harmonic Analysis VolII Die Grundlehren der mathematischen WissenschaftenSpringer Berlin Germany 1970

[26] H G Feichtinger and T Werther ldquoRobustness of regularsampling in Sobolev algebrasrdquo in Sampling Wavelets andTomography J Benedetto and A I Zayed Eds pp 83ndash113Birkhauser Boston Boston Mass USA 2004

[27] E Popescu ldquoNonlocal Dirichlet forms generated by pseudod-ifferential operators on compact abelian groupsrdquo Proceedingsin Applied Mathematics and Mechanics vol 7 pp 2160001ndash2160002 2007

[28] A Weil LrsquoIntegration dans les Groupes Topologiques et ses Appli-cations Hermann et Cie Paris France 1940

[29] H Triebel ldquoFunction spaces in Lipschitz domains and onLipschitz manifolds Characteristic functions as pointwise mul-tipliersrdquo Revista Matematica Complutense vol 15 no 2 pp475ndash524 2002

[30] P Gorka ldquoPego theorem on locally compact abelian groupsrdquoJournal of Algebra and Its Applications vol 13 no 4 Article ID1350143 2014

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

2 Journal of Function Spaces

group 119866 which limits its applicability In this paper weremove this assumption and prove a version of the Rellichlemma which can be applied in great generality and we alsoimprove our original Rellich-Kondrachov theorem provenin [1] Moreover we introduce Sobolev spaces on subsets ofLCA groups in analogy with the Sobolev spaces defined ondomains ofR119899 As in this classical case we expect these spacesto be useful in the study of differential equations and otherapplications [23]

We organize this paper as follows In Section 2 we recallour definition of Sobolev spaces and our previous embeddingand compactness results In Section 3 we state and proveour new compactness results and in Section 4 we discussSobolev spaces on subsets of LCA groups

We use standard notations from harmonic analysis [2425] Let us fix a locally compact abelian group 119866 We denoteby 119889119909 the uniqueHaarmeasure of119866 and by119866and the dual groupof the group119866 that is119866and is the locally compact abelian groupof all continuous group homomorphisms from119866 to the circlegroup 119879 The 119871119901 spaces over 119866 are defined as usual

119871119901(119866) = 119891 119866 997888rarr C int

119866

1003816100381610038161003816119891 (119909)1003816100381610038161003816

119901119889119909 lt infin (3)

and we set

10038171003817100381710038171198911003817100381710038171003817119871119901(119866)

= (int119866

1003816100381610038161003816119891 (119909)1003816100381610038161003816

119901119889119909)

1119901

(4)

We also recall that the Fourier transform on 119866 is definedas follows if 119891 isin 119871

1(119866) then its Fourier transform is the

function 119891 119866andrarr C given by

119891 (120585) = int119866

120585(119909)119891 (119909) 119889119909 (5)

We consider general LCA groups in Section 2 but werestrict ourselves to compact abelian groups when provingcompactness results in Section 3

2 Sobolev Spaces

We introduce Sobolev spaces following our previous papers[1 2]Our definition uses essentially the Fourier transform forLCA groups and as explained in [1] it generalizes naturallythe standard notions of Sobolev spaces in the case of T119899 andR119899 see [26] and [4 Chapter 4]

We denote by Γ the set

Γ = 120574 119866and997888rarr [0infin) exist119888120574

forall120572120573isin119866and120574 (120572120573)

le 119888120574 [120574 (120572) + 120574 (120573)]

(6)

Definition 1 Let us fix a map 120574 isin Γ and a nonnegative realnumber 119904 We will say that 119891 isin 119871

2(119866) belongs to the Sobolev

space119867119904

120574(119866) if the following integral is finite

int119866and

(1 + 120574(120585)2)11990410038161003816100381610038161003816119891 (120585)

10038161003816100381610038161003816

2

119889120585 (7)

Moreover for 119891 isin 119867119904

120574(119866) its norm 119891

119867119904120574(119866)

is defined asfollows

10038171003817100381710038171198911003817100381710038171003817119867119904120574(119866)

= (int119866and

(1 + 120574(120585)2)11990410038161003816100381610038161003816119891 (120585)

10038161003816100381610038161003816

2

119889120585)

12

(8)

Remark 2 Aparticular instance ofDefinition 1 appears in thepaper [26] by Feichtinger and Werther Another particularcase of our definition is in [27] We also note that in 119901-adicanalysis Sobolev spaces are defined in a way analogous to ourDefinition 1 if we take 120574(120585) = 120585119901 where sdot 119901 is a 119901-adicnorm on Q119899

119901≃ Q119899and

119901 then (7) and (8) allow us to recover the

119901-adic Sobolev spaces considered in [11]

Remark 3 The fact that 120574 isin Γ implies that our spaces119867119904

120574(119866)

are Banach algebras under some assumptions on 119904 see ourprevious paper [1]

We recall the following results proven in [1]

Proposition 4 Let 119866 be a locally compact abelian group Onehas the following

(1) Consider 119867119904

120574(119866) 997893rarr 119871

2(119866) Moreover for each 119891 isin

119867119904

120574(119866) the following inequality holds

100381710038171003817100381711989110038171003817100381710038171198712(119866)

le1003817100381710038171003817119891

1003817100381710038171003817119867119904120574(119866)

(9)

(2) If 119904 gt 120590 then 119867119904

120574(119866) 997893rarr 119867

120590

120574(119866) Moreover the

following inequality holds1003817100381710038171003817119891

1003817100381710038171003817119867120590120574(119866)

le1003817100381710038171003817119891

1003817100381710038171003817119867119904120574(119866)

(10)

Theorem 5 Let119866 be a locally compact abelian group One hasthe following

(1) If 1(1 + 120574(sdot)2) isin 119871

119904(119866

and) then

119867119904

120574(119866) 997893rarr 119862 (119866) (11)

inwhich119862(119866) denotes the space of continuous complex-valued functions on119866Moreover there exists a constant119862(120574 119904) such that for each 119891 isin 119867

119904

120574(119866) the following

inequality holds1003817100381710038171003817119891

1003817100381710038171003817119862(119866)le 119862 (120574 119904)

10038171003817100381710038171198911003817100381710038171003817119867119904120574(119866)

(12)

(2) If 120572 gt 119904 and 1(1 + 120574(sdot)2) isin 119871

120572(119866

and) then

119867119904

120574(119866) 997893rarr 119871

120572lowast

(119866) (13)

where120572lowast= 2120572(120572minus119904)Moreover there exists a constant

119863(120574 119904) such that for each 119891 isin 119867119904

120574(119866) the following

inequality holds1003817100381710038171003817119891

1003817100381710038171003817119871120572lowast(119866)

le 119863 (120574 119904)1003817100381710038171003817119891

1003817100381710038171003817119867119904120574(119866)

(14)

Theorem 5 is our version of the classical Sobolev embed-ding theorem appearing for instance in [3] for the case 119866 =

R119899

Journal of Function Spaces 3

3 Compact Embedding

We recall that the notation 119860 997893rarr997893rarr 119861 means that the space119860 is compactly embedded in 119861 We begin with our refinedversion of the classical Rellich lemma

Theorem 6 Let 119866 be a compact group If lim120585rarrinfin120574(120585) = infinthat is for each119872 gt 0 there exists finite set119880 such that for any120585 isin 119866

and 119880 one has 120574(120585) ge 119872 then for all 119904 gt 120575

119867119904

120574(119866) 997893rarr997893rarr 119867

120575

120574(119866) (15)

Proof We begin the proof stating a classical fact (see eg[28]) on the characterization of precompact sets in 119871

119901(119867)

spaces

Theorem7 (see [28]) LetΦ be a family of functions in 119871119901(119867)

1 le 119901 lt infin Then Φ is compact in 119871119901(119867) if and only if the

following conditions hold

(i) There exists 119872 gt 0 such that 120601119871119901(119867)

le 119872 for all120601 isin Φ

(ii) For every 120576 gt 0 there exists compact set 119870 in 119866 suchthat for each 120601 isin Φ

10038171003817100381710038171206011003817100381710038171003817119871119901(119867119870)

le 120576 (16)

(iii) For all 120576 gt 0 there exists unit neighborhood 119881 suchthat for all 119909 isin 119881 and all 120601 isin Φ

1003817100381710038171003817119871119909120601 minus 1206011003817100381710038171003817119871119901(119867)

le 120576 (17)

Now we return to the proof of Theorem 6 Let 119891119899 be abounded sequence in 119867

119904

120574(119866) 119891119899119867119904

120574(119866)

le 119872 We need toshow that there exists subsequence that converges strongly in119867

120575

120574(119866) We will prove this fact by showing that the following

sequence

119892119899 (120585) =10038161003816100381610038161003816119891119899 (120585)

10038161003816100381610038161003816(1 + 120574

2(120585))

1205752 (18)

is compact in 1198712(119866

and) We use Weilrsquos theorem since 120575 lt 119904 we

get

1003817100381710038171003817119892119899

10038171003817100381710038171198712(119866and)le1003817100381710038171003817119891119899

1003817100381710038171003817119867119904120574(119866)

le 119872 (19)

and so (i) holds Now we consider condition (ii) Let us fix120598 gt 0 We consider two cases if 119866and is finite group then wecan take simply 119870 = 119866

and and condition (ii) is satisfied Onthe other hand if the dual group 119866

and is infinite it is enoughto recall that if 119866 is a compact group then by the Pontryaginduality theorem its dual 119866and is discrete and therefore everycompact set must be finite From our assumption we can finda compact set 119870 such that

1

(1 + 1205742 (120585))119904minus120575

le120598

1198722 for all 120585 isin 119866

and 119870 (20)

Hence

int119866and119870

10038161003816100381610038161003816119891119899 (120585)

10038161003816100381610038161003816

2

(1 + 1205742(120585))

120575

119889120585

= int119866and119870

10038161003816100381610038161003816119891119899 (120585)

10038161003816100381610038161003816

2 (1 + 1205742(120585))

119904

(1 + 1205742 (120585))119904minus120575

119889120585

le sup120585isin119866and119870

1

(1 + 1205742 (120585))119904minus120575

int119866and119870

10038161003816100381610038161003816119891119899 (120585)

10038161003816100381610038161003816

2

(1 + 1205742(120585))

119904

119889120585

le1003817100381710038171003817119891119899

1003817100381710038171003817

2

119867119904120574(119866)

sup120585isin119866and119870

1

(1 + 1205742 (120585))119904minus120575

le 120598

(21)

and so (ii) holds It remains to check condition (iii) Since 119866and

is discrete and each set is open we can take 119881 = 119890 where 119890is unit in 119866

and Thus condition (iii) is satisfied and Theorem 6follows fromWeilrsquos result

Theorem 6 appears in our previous paper [2] under theadditional assumption that the dual group 119866and is countable

Now we consider embeddings of 119867119904

120574(119866) into 119862(119866) and

119871119901(119866)We proved in [1] that119867119904

120574(119866) is continuously embedded

in 119862(119866) We prove in Theorem 10 below that if 119866 is compactthen 119867

119904

120574997893rarr997893rarr 119862(119866) and finally in Theorem 11 we consider

a version of the Rellich-Kondrachov which refines an analo-gous result from [1] We need the following lemma

Lemma 8 Let 119867 be a discrete group and 119891 isin 1198711(119867) Then

for every 120576 gt 0 there exists a finite set 119880 such that for anyℎ isin 119867 119880 one has |119891(ℎ)| le 120576

Proof Let us suppose that the theorem is not true so thatthere is a number 120576 gt 0 such that for every finite set 119880there exists ℎ isin 119867 119880 with |119891(ℎ)| ge 120576 Let 1198801 be a finiteset and let ℎ1 isin 119867 1198801 be such that |119891(ℎ1)| ge 120576 We define1198802 = 1198801cupℎ1 Since1198802 is finite there exists ℎ2 isin 1198671198802 suchthat |119891(ℎ2)| ge 120576 By induction we get a sequence of sets 119880119894

and ℎ119894 for 119894 = 1 2 such that119880119894+1 = 119880119894 cup ℎ119894 and for each119894 we have |119891(ℎ119894)| ge 120576 Since Haar measure on discrete groupsis a multiple of the counting measure we get

int119867

1003816100381610038161003816119891 (ℎ)1003816100381610038161003816 119889ℎ = int

119867119880

1003816100381610038161003816119891 (ℎ)1003816100381610038161003816 119889ℎ + int

119880

1003816100381610038161003816119891 (ℎ)1003816100381610038161003816 119889ℎ

ge

infin

sum

119894=1

1003816100381610038161003816119891 (ℎ119894)1003816100381610038161003816 ge

infin

sum

119894=1

120576 = +infin

(22)

where 119880 = ⋃infin

119894=1119880119894 and this is a contradiction

Remark 9 We note that if 119867 is countable the proof ofLemma 8 is elementary int

119867|119891(ℎ)|119889ℎ = 119886 sum

infin

119894=1|119891(ℎ119894)| for

some 119886 isin C and suminfin

119894=1|119891(ℎ119894)| lt infin implies that given 120598 gt 0

there exists 119870 gt 0 such that suminfin

119894=119870|119891(ℎ119894)| lt 120598 The result then

follows

Theorem10 If119866 is compact 1(1+1205742(sdot)) isin 119871120572(119866

and) and 119904 lt 120572

then119867

119904

120574(119866) 997893rarr997893rarr 119862 (119866) (23)

4 Journal of Function Spaces

Proof Using Lemma 8 for the function 1(1 + 1205742(sdot))

120572 we seethat 120574 satisfies the assumptions ofTheorem 6 so for 119904 lt 120572 weget that

119867119904

120574(119866) 997893rarr997893rarr 119867

120572

120574(119866) (24)

Moreover thanks to the first part of Theorem 5 we have

119867120572

120574(119866) 997893rarr 119862 (119866) (25)

and the proof follows

Theorem11 If119866 is compact 1(1+1205742(sdot)) isin 119871120572(119866

and) and 119904 lt 120572

then

119867119904

120574(119866) 997893rarr997893rarr 119871

119901(119866) (26)

for all 119901 lt 120572lowast= 2120572(120572 minus 119904)

Proof Let us take 120575 = 120572 minus 2120572119901 then 120575 lt 119904 Thus byTheorem 6 we have the compact embedding

119867119904

120574(119866) 997893rarr997893rarr 119867

120575

120574(119866) (27)

Next using the second part of Theorem 5 we have thecontinuous embedding

119867120575

120574(119866) 997893rarr 119871

120572lowast

120575 (119866) (28)

where 120572lowast

120575= 2120572(120572 minus 120575) = 119901

4 Sobolev Spaces on Subsets of LCA Groups

In this section we deal with Sobolev spaces defined onsubsets of locally compact abelian groups As mentioned inSection 1 we are motivated by analogous studies of functionspaces on domains of R119899 (see eg [29]) by the fact thatinteresting applications exist [23] and by the possibility ofusing them as tools for the study of differential equations onsubsets of LCA groupsWe start with the following definition

Definition 12 Let119880 be a subset of a LCA group 119866 We definethe Sobolev space 119867

119904

120574(119880) sub 119871

2(119880) as the space of all 119891 isin

1198712(119880) such that there exists 119865 isin 119867

119904

120574(119866) with 119865|119880 = 119891 and

we equip it with the norm

10038171003817100381710038171198911003817100381710038171003817119867119904120574(119880)

= inf 119865119867119904120574(119866) 119865 isin 119867

119904

120574(119866) and 119865|119880 = 119891

(29)

An analogous definition (of spaces119861119904

119901119902on domains ofR119899)

appears in [29] see his Definition 23 It can be easily shownthat 119867119904

120574(119880) is a Banach space We will say that it is a local

Sobolev spaceUsing appropriate embeddings for119867119904

120574(119866) and the defini-

tion of119867119904

120574(119880) we can prove the following

Theorem 13 Let 119866 be a locally compact abelian group and let119880 sub 119866 Then we have

(1) The continuous inclusion 119867119904

120574(119880) 997893rarr 119871

2(119880) holds

Moreover for each 119891 isin 119867119904

120574(119880) the following inequality

holds1003817100381710038171003817119891

10038171003817100381710038171198712(119880)le1003817100381710038171003817119891

1003817100381710038171003817119867119904120574(119880)

(30)

(2) If 119904 gt 120590 then119867119904

120574(119866) 997893rarr 119867

120590

120574(119880) Moreover the follow-

ing inequality holds1003817100381710038171003817119891

1003817100381710038171003817119867120590120574(119880)

le1003817100381710038171003817119891

1003817100381710038171003817119867119904120574(119880)

(31)

(3) If 1(1 + 120574(sdot)2) isin 119871

119904(119866

and) then 119867

119904

120574(119880) 997893rarr 119862(119880)

Moreover there exists a constant 119862(120574 119904) such that foreach 119891 isin 119867

119904

120574(119880) the following inequality holds

10038171003817100381710038171198911003817100381710038171003817119862(119880)

le 119862 (120574 119904)1003817100381710038171003817119891

1003817100381710038171003817119867119904120574(119880)

(32)

(4) If 120572 gt 119904 and 1(1 + 120574(sdot)2) isin 119871

120572(119866

and) then 119867

119904

120574(119880) 997893rarr

119871120572lowast

(119880) where 120572lowast= 2120572(120572 minus 119904) Moreover there exists

a constant 119863(120574 119904) such that for each 119891 isin 119867119904

120574(119866) the

following inequality holds1003817100381710038171003817119891

1003817100381710038171003817119871120572lowast(119880)

le 119863 (120574 119904)1003817100381710038171003817119891

1003817100381710038171003817119867119904120574(119880)

(33)

We now prove the following compactness theorem indetail

Theorem 14 Let119866 be an LCA group and let119880 be a subset of119866of finite measure Assume that 1(1 + 1205742(sdot)) isin 119871

120572(119866

and) for some

120572 gt 119904 and that1003816100381610038161003816120585 (ℎ) minus 1

1003816100381610038161003816

(1 + 1205742 (120585))119904 997888rarr

ℎrarr1198900 119906119899119894119891119900119903119898119897119910 119908119894119905ℎ 119903119890119904119901119890119888119905 119905119900 120585 isin 119866

and

(34)

Then for all 119901 lt 120572lowast one has the compact embedding

119867119904

120574(119880) 997893rarr997893rarr 119871

119901(119880) (35)

The convergence concept used in (34) is explained in ourprevious paper [1] Let us mention that a condition similarto (34) appears in the characterization of precompact sets in1198712(119866) via Fourier transform see [30]

Proof We will need two lemmas which we proved in [1]

Lemma 15 Let 119865 isin 119867119904

120574(119866) and assume that

1003816100381610038161003816120585 (ℎ) minus 11003816100381610038161003816

(1 + 1205742 (120585))119904 997888rarr

ℎrarr1198900 119906119899119894119891119900119903119898119897119910 119908119894119905ℎ 119903119890119904119901119890119888119905 119905119900 120585 isin 119866

and

(36)

Then for every ℎ isin 119866

int119866

|119865 (119909ℎ) minus 119865 (119909)|2119889119909 le 119862 (ℎ) 119865

2

119867119904120574(119866)

(37)

where 119862(ℎ) rarrℎrarr119890

0

Journal of Function Spaces 5

Lemma 16 Let us denote by I the set of all symmetric unitneighbourhoods of group G partially ordered by inclusion Let(120601119881)119881isinI be a Dirac net and let 119865 isin 119867

119904

120574(119866) Then

int119866

1003816100381610038161003816119865 lowast 120601119881 (119909) minus 119865 (119909)1003816100381610038161003816

2119889119909 le 119865119867119904

120574(119866)sup

119910isin119881

119862 (119910) (38)

Nowwe can continuewith the proof ofTheorem 14 Let119891119899

be a bounded sequence in119867119904

120574(119880) that is 119891119899119867119904

120574(119880)

le 119862 Fromthe very definition there exist functions 119865119899 such that 119865119899 = 119891119899

in119880119865119899 isin 119867119904

120574(119866) and 119865119899119867119904

120574(119866)

le 119891119899119867119904120574(119880)

+1 for 119899 = 1 2 Now by Theorem 13 we get that the sequence 119891119899 is boundedin 119871

120572lowast

(119880) Hence there exists a subsequence 119891119899119896such that

119891119899119896 119891 in 119871

120572lowast

(119880) and there also exists a subsequence 119865119899119896

such that 119865119899119896 119865 in 119871

120572lowast

(119866) For simplicity we simply write119891119899 and 119865119899 for these subsequences From the basic propertiesof weak convergence we have that 119865 = 119891 in 119880 We will showthat for 119902 lt 120572

lowast we have

119891119899 997888rarr 119891 in 119871119902(119880) (39)

Let us set 119865(119881) = 119865 lowast 120601119881 From the triangle inequality we get

1003817100381710038171003817119891119899 minus 11989110038171003817100381710038171198712(119880)

le1003817100381710038171003817119891119899 minus 119865119899(119881)

10038171003817100381710038171198712(119880)

+1003817100381710038171003817119865119899(119881) minus 119865(119881)

10038171003817100381710038171198712(119880)+1003817100381710038171003817119865(119881) minus 119891

10038171003817100381710038171198712(119880)

le1003817100381710038171003817119865119899 minus 119865119899(119881)

10038171003817100381710038171198712(119866)+1003817100381710038171003817119865119899(119881) minus 119865(119881)

10038171003817100381710038171198712(119880)

+1003817100381710038171003817119865(119881) minus 119865

10038171003817100381710038171198712(119866)

(40)

From Lemma 15 we get

sup119899

int119866

10038161003816100381610038161003816119865119899(119881)

(119909) minus 119865119899 (119909)10038161003816100381610038161003816

2

119889119909 le sup119899

1003817100381710038171003817119865119899

1003817100381710038171003817

2

119867119904120574(119866)

sup119910isin119881

119862 (119910)

le (119862 + 1)2sup119910isin119881

119862 (119910)

(41)

Furthermore it can be easily shown that for each 120598 gt 0 thereexists 119881120598 such that for all 119881 isin I with 119881 sub 119881120598 we have

1003817100381710038171003817119865(119881) minus 11986510038171003817100381710038171198712(119866)

le120598

3 (42)

Hance we can choose 119881120598 such that

1003817100381710038171003817119891119899 minus 11989110038171003817100381710038171198712(119880)

le2

3120598 +

10038171003817100381710038171003817119865119899(119881120598)

minus 119865(119881120598)

100381710038171003817100381710038171198712(119880) (43)

We need to estimate 119865119899(119881120598)minus 119865(119881120598)

1198712(119880)

Since 119865119899 119865 in119871120572lowast

(119866) we get that 119865119899(119881120598)(119909) rarr

119899rarrinfin119865(119881120598)

(119909) Moreover wehave

10038161003816100381610038161003816119865119899(119881120598)

(119909) minus 119865(119881120598)(119909)

10038161003816100381610038161003816

2

=

10038161003816100381610038161003816100381610038161003816

int119866

(119865119899(119910) minus 119865(119910))120601119881120598(119910

minus1119909)119889119910

10038161003816100381610038161003816100381610038161003816

2

=

10038161003816100381610038161003816100381610038161003816

int119866

(119865119899(119910) minus 119865(119910))radic120601119881120598(119910minus1119909)radic120601119881120598

(119910minus1119909)119889119910

10038161003816100381610038161003816100381610038161003816

2

le int119866

120601119881120598(119910

minus1119909) 119889120583119866 (119910)

times int119866

1003816100381610038161003816119865119899 (119910) minus 119865 (119910)1003816100381610038161003816

2120601119881120598

(119910minus1119909) 119889119910

le sup119911isin119881120598

120601119881120598(119911) int

119866

1003816100381610038161003816119865119899 (119910) minus 119865 (119910)1003816100381610038161003816

2120601119881120598

(119910minus1119909) 119889119910

(44)

Hence since 119880 has a finite measure we get from Lebesgueconvergence theorem that

10038171003817100381710038171003817119865119899(119881120598)

minus 119865(119881120598)

100381710038171003817100381710038171198712(119880)le120598

3 (45)

We conclude that 119891119899 rarr 119891 in 1198712(119880) In order to finish the

proof it is enough to use the Vitali convergence theoremWeconclude that 119891119899 rarr 119891 in 119871

119902(119880) for 119902 lt 120572

lowast

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

Przemysław Gorkarsquos research is partially supported by theEuropean Union in the framework of European Social Fundthrough the Warsaw University of Technology DevelopmentProgram EnriqueG Reyesrsquo research is partially supported bythe Project FONDECYT no 1111042

References

[1] P Gorka and E G Reyes ldquoSobolev spaceson locally compactabelian groupsrdquo httparXivorgabs12083053 in press

[2] P Gorka T Kostrzewa and E G Reyes ldquoThe Rellich lemmaon compact abelian groups and equations of infinite orderrdquoInternational Journal of Geometric Methods in Modern Physicsvol 10 no 2 Article ID 1220030 11 pages 2013

[3] R A Adams Sobolev Spaces Academic Press New York NYUSA 1975

[4] M E Taylor Partial Differential Equations vol 23 SpringerNew York NY USA 1996

[5] T Aubin Some Nonlinear Problems in Riemannian GeometrySpringer Berlin Germany 1998

[6] E Hebey Sobolev Spaces on Riemannian Manifolds vol 1635Springer Berlin Germany 1996

[7] P Hajłasz ldquoSobolev spaces on an arbitrary metric spacerdquoPotential Analysis vol 5 no 4 pp 403ndash415 1996

[8] P Hajłasz and P Koskela ldquoSobolev met Poincarerdquo Memoirs ofthe American Mathematical Society vol 145 no 688 2000

[9] N Shanmugalingam ldquoNewtonian spaces an extension ofSobolev spaces to metric measure spacesrdquo Revista MatematicaIberoamericana vol 16 no 2 pp 243ndash279 2000

6 Journal of Function Spaces

[10] Y-C Kim ldquoA simple proof of the 119901-adic version of the Sobolevembedding theoremrdquo Korean Mathematical Society vol 25 no1 pp 27ndash36 2010

[11] J J Rodrıguez-Vega and W A Zuniga-Galindo ldquoElliptic pseu-dodifferential equations and Sobolev spaces over 119901-adic fieldsrdquoPacific Journal ofMathematics vol 246 no 2 pp 407ndash420 2010

[12] H Bahouri C Fermanian-Kammerer and I Gallagher ldquoAnal-yse de lrsquoespace des phases et calcul pseudo-differential sur legroupe de Heisenbergrdquo Comptes Rendus Mathematique vol347 no 17-18 pp 1021ndash1024 2009

[13] P Gorka H Prado and E G Reyes ldquoNonlinear equations withinfinitely many derivativesrdquo Complex Analysis and OperatorTheory vol 5 no 1 pp 313ndash323 2011

[14] P Gorka H Prado and E G Reyes ldquoGeneralized euclideanbosonic string equationsrdquo in Operator Theory Advances andApplications vol 224 pp 147ndash169 Springer Basel Switzerland2012

[15] P Gorka H Prado and E G Reyes ldquoOn a general class ofnonlocal equationsrdquo Annales Henri Poincare vol 14 no 4 pp947ndash966 2013

[16] P Gorka H Prado and E G Reyes ldquoFunctional calculusvia Laplace transform and equations with infinitely manyderivativesrdquo Journal of Mathematical Physics vol 51 no 10Article ID 103512 2010

[17] P Gorka H Prado and E G Reyes ldquoThe initial value problemfor ordinary differential equations with infinitely many deriva-tivesrdquo Classical and Quantum Gravity vol 29 no 6 Article ID065017 2012

[18] N Barnaby and N Kamran ldquoDynamics with infinitely manyderivatives the initial value problemrdquo Journal of High EnergyPhysics no 2 40 pages 2008

[19] G Calcagni M Montobbio and G Nardelli ldquoLocalization ofnonlocal theoriesrdquo Physics Letters B vol 662 no 3 pp 285ndash2892008

[20] V S Vladimirov ldquoOn the equation of a 119901-adic open string for ascalar tachyon fieldrdquo Izvestiya vol 69 no 3 pp 487ndash512 2005

[21] E Witten ldquoNoncommutative geometry and string field theoryrdquoNuclear Physics B vol 268 no 2 pp 253ndash294 1986

[22] D Zhou and C J C Burges ldquoHigh-order regularization ongraphsrdquo in Proceedings of the 6th International Workshop onMining and Learning with Graphs Helsinki Finland 2008httpwwwcishutfiMLG08paperspostersessionpaper39pdf

[23] H Triebel ldquoWavelets in function spaces on Lipschitz domainsrdquoMathematische Nachrichten vol 280 no 9-10 pp 1205ndash12182007

[24] E Hewitt andK A RossAbstract Harmonic Analysis Vol I DieGrundlehren der mathematischen Wissenschaften SpringerBerlin Germany 1963

[25] E Hewitt and K A Ross Abstract Harmonic Analysis VolII Die Grundlehren der mathematischen WissenschaftenSpringer Berlin Germany 1970

[26] H G Feichtinger and T Werther ldquoRobustness of regularsampling in Sobolev algebrasrdquo in Sampling Wavelets andTomography J Benedetto and A I Zayed Eds pp 83ndash113Birkhauser Boston Boston Mass USA 2004

[27] E Popescu ldquoNonlocal Dirichlet forms generated by pseudod-ifferential operators on compact abelian groupsrdquo Proceedingsin Applied Mathematics and Mechanics vol 7 pp 2160001ndash2160002 2007

[28] A Weil LrsquoIntegration dans les Groupes Topologiques et ses Appli-cations Hermann et Cie Paris France 1940

[29] H Triebel ldquoFunction spaces in Lipschitz domains and onLipschitz manifolds Characteristic functions as pointwise mul-tipliersrdquo Revista Matematica Complutense vol 15 no 2 pp475ndash524 2002

[30] P Gorka ldquoPego theorem on locally compact abelian groupsrdquoJournal of Algebra and Its Applications vol 13 no 4 Article ID1350143 2014

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Journal of Function Spaces 3

3 Compact Embedding

We recall that the notation 119860 997893rarr997893rarr 119861 means that the space119860 is compactly embedded in 119861 We begin with our refinedversion of the classical Rellich lemma

Theorem 6 Let 119866 be a compact group If lim120585rarrinfin120574(120585) = infinthat is for each119872 gt 0 there exists finite set119880 such that for any120585 isin 119866

and 119880 one has 120574(120585) ge 119872 then for all 119904 gt 120575

119867119904

120574(119866) 997893rarr997893rarr 119867

120575

120574(119866) (15)

Proof We begin the proof stating a classical fact (see eg[28]) on the characterization of precompact sets in 119871

119901(119867)

spaces

Theorem7 (see [28]) LetΦ be a family of functions in 119871119901(119867)

1 le 119901 lt infin Then Φ is compact in 119871119901(119867) if and only if the

following conditions hold

(i) There exists 119872 gt 0 such that 120601119871119901(119867)

le 119872 for all120601 isin Φ

(ii) For every 120576 gt 0 there exists compact set 119870 in 119866 suchthat for each 120601 isin Φ

10038171003817100381710038171206011003817100381710038171003817119871119901(119867119870)

le 120576 (16)

(iii) For all 120576 gt 0 there exists unit neighborhood 119881 suchthat for all 119909 isin 119881 and all 120601 isin Φ

1003817100381710038171003817119871119909120601 minus 1206011003817100381710038171003817119871119901(119867)

le 120576 (17)

Now we return to the proof of Theorem 6 Let 119891119899 be abounded sequence in 119867

119904

120574(119866) 119891119899119867119904

120574(119866)

le 119872 We need toshow that there exists subsequence that converges strongly in119867

120575

120574(119866) We will prove this fact by showing that the following

sequence

119892119899 (120585) =10038161003816100381610038161003816119891119899 (120585)

10038161003816100381610038161003816(1 + 120574

2(120585))

1205752 (18)

is compact in 1198712(119866

and) We use Weilrsquos theorem since 120575 lt 119904 we

get

1003817100381710038171003817119892119899

10038171003817100381710038171198712(119866and)le1003817100381710038171003817119891119899

1003817100381710038171003817119867119904120574(119866)

le 119872 (19)

and so (i) holds Now we consider condition (ii) Let us fix120598 gt 0 We consider two cases if 119866and is finite group then wecan take simply 119870 = 119866

and and condition (ii) is satisfied Onthe other hand if the dual group 119866

and is infinite it is enoughto recall that if 119866 is a compact group then by the Pontryaginduality theorem its dual 119866and is discrete and therefore everycompact set must be finite From our assumption we can finda compact set 119870 such that

1

(1 + 1205742 (120585))119904minus120575

le120598

1198722 for all 120585 isin 119866

and 119870 (20)

Hence

int119866and119870

10038161003816100381610038161003816119891119899 (120585)

10038161003816100381610038161003816

2

(1 + 1205742(120585))

120575

119889120585

= int119866and119870

10038161003816100381610038161003816119891119899 (120585)

10038161003816100381610038161003816

2 (1 + 1205742(120585))

119904

(1 + 1205742 (120585))119904minus120575

119889120585

le sup120585isin119866and119870

1

(1 + 1205742 (120585))119904minus120575

int119866and119870

10038161003816100381610038161003816119891119899 (120585)

10038161003816100381610038161003816

2

(1 + 1205742(120585))

119904

119889120585

le1003817100381710038171003817119891119899

1003817100381710038171003817

2

119867119904120574(119866)

sup120585isin119866and119870

1

(1 + 1205742 (120585))119904minus120575

le 120598

(21)

and so (ii) holds It remains to check condition (iii) Since 119866and

is discrete and each set is open we can take 119881 = 119890 where 119890is unit in 119866

and Thus condition (iii) is satisfied and Theorem 6follows fromWeilrsquos result

Theorem 6 appears in our previous paper [2] under theadditional assumption that the dual group 119866and is countable

Now we consider embeddings of 119867119904

120574(119866) into 119862(119866) and

119871119901(119866)We proved in [1] that119867119904

120574(119866) is continuously embedded

in 119862(119866) We prove in Theorem 10 below that if 119866 is compactthen 119867

119904

120574997893rarr997893rarr 119862(119866) and finally in Theorem 11 we consider

a version of the Rellich-Kondrachov which refines an analo-gous result from [1] We need the following lemma

Lemma 8 Let 119867 be a discrete group and 119891 isin 1198711(119867) Then

for every 120576 gt 0 there exists a finite set 119880 such that for anyℎ isin 119867 119880 one has |119891(ℎ)| le 120576

Proof Let us suppose that the theorem is not true so thatthere is a number 120576 gt 0 such that for every finite set 119880there exists ℎ isin 119867 119880 with |119891(ℎ)| ge 120576 Let 1198801 be a finiteset and let ℎ1 isin 119867 1198801 be such that |119891(ℎ1)| ge 120576 We define1198802 = 1198801cupℎ1 Since1198802 is finite there exists ℎ2 isin 1198671198802 suchthat |119891(ℎ2)| ge 120576 By induction we get a sequence of sets 119880119894

and ℎ119894 for 119894 = 1 2 such that119880119894+1 = 119880119894 cup ℎ119894 and for each119894 we have |119891(ℎ119894)| ge 120576 Since Haar measure on discrete groupsis a multiple of the counting measure we get

int119867

1003816100381610038161003816119891 (ℎ)1003816100381610038161003816 119889ℎ = int

119867119880

1003816100381610038161003816119891 (ℎ)1003816100381610038161003816 119889ℎ + int

119880

1003816100381610038161003816119891 (ℎ)1003816100381610038161003816 119889ℎ

ge

infin

sum

119894=1

1003816100381610038161003816119891 (ℎ119894)1003816100381610038161003816 ge

infin

sum

119894=1

120576 = +infin

(22)

where 119880 = ⋃infin

119894=1119880119894 and this is a contradiction

Remark 9 We note that if 119867 is countable the proof ofLemma 8 is elementary int

119867|119891(ℎ)|119889ℎ = 119886 sum

infin

119894=1|119891(ℎ119894)| for

some 119886 isin C and suminfin

119894=1|119891(ℎ119894)| lt infin implies that given 120598 gt 0

there exists 119870 gt 0 such that suminfin

119894=119870|119891(ℎ119894)| lt 120598 The result then

follows

Theorem10 If119866 is compact 1(1+1205742(sdot)) isin 119871120572(119866

and) and 119904 lt 120572

then119867

119904

120574(119866) 997893rarr997893rarr 119862 (119866) (23)

4 Journal of Function Spaces

Proof Using Lemma 8 for the function 1(1 + 1205742(sdot))

120572 we seethat 120574 satisfies the assumptions ofTheorem 6 so for 119904 lt 120572 weget that

119867119904

120574(119866) 997893rarr997893rarr 119867

120572

120574(119866) (24)

Moreover thanks to the first part of Theorem 5 we have

119867120572

120574(119866) 997893rarr 119862 (119866) (25)

and the proof follows

Theorem11 If119866 is compact 1(1+1205742(sdot)) isin 119871120572(119866

and) and 119904 lt 120572

then

119867119904

120574(119866) 997893rarr997893rarr 119871

119901(119866) (26)

for all 119901 lt 120572lowast= 2120572(120572 minus 119904)

Proof Let us take 120575 = 120572 minus 2120572119901 then 120575 lt 119904 Thus byTheorem 6 we have the compact embedding

119867119904

120574(119866) 997893rarr997893rarr 119867

120575

120574(119866) (27)

Next using the second part of Theorem 5 we have thecontinuous embedding

119867120575

120574(119866) 997893rarr 119871

120572lowast

120575 (119866) (28)

where 120572lowast

120575= 2120572(120572 minus 120575) = 119901

4 Sobolev Spaces on Subsets of LCA Groups

In this section we deal with Sobolev spaces defined onsubsets of locally compact abelian groups As mentioned inSection 1 we are motivated by analogous studies of functionspaces on domains of R119899 (see eg [29]) by the fact thatinteresting applications exist [23] and by the possibility ofusing them as tools for the study of differential equations onsubsets of LCA groupsWe start with the following definition

Definition 12 Let119880 be a subset of a LCA group 119866 We definethe Sobolev space 119867

119904

120574(119880) sub 119871

2(119880) as the space of all 119891 isin

1198712(119880) such that there exists 119865 isin 119867

119904

120574(119866) with 119865|119880 = 119891 and

we equip it with the norm

10038171003817100381710038171198911003817100381710038171003817119867119904120574(119880)

= inf 119865119867119904120574(119866) 119865 isin 119867

119904

120574(119866) and 119865|119880 = 119891

(29)

An analogous definition (of spaces119861119904

119901119902on domains ofR119899)

appears in [29] see his Definition 23 It can be easily shownthat 119867119904

120574(119880) is a Banach space We will say that it is a local

Sobolev spaceUsing appropriate embeddings for119867119904

120574(119866) and the defini-

tion of119867119904

120574(119880) we can prove the following

Theorem 13 Let 119866 be a locally compact abelian group and let119880 sub 119866 Then we have

(1) The continuous inclusion 119867119904

120574(119880) 997893rarr 119871

2(119880) holds

Moreover for each 119891 isin 119867119904

120574(119880) the following inequality

holds1003817100381710038171003817119891

10038171003817100381710038171198712(119880)le1003817100381710038171003817119891

1003817100381710038171003817119867119904120574(119880)

(30)

(2) If 119904 gt 120590 then119867119904

120574(119866) 997893rarr 119867

120590

120574(119880) Moreover the follow-

ing inequality holds1003817100381710038171003817119891

1003817100381710038171003817119867120590120574(119880)

le1003817100381710038171003817119891

1003817100381710038171003817119867119904120574(119880)

(31)

(3) If 1(1 + 120574(sdot)2) isin 119871

119904(119866

and) then 119867

119904

120574(119880) 997893rarr 119862(119880)

Moreover there exists a constant 119862(120574 119904) such that foreach 119891 isin 119867

119904

120574(119880) the following inequality holds

10038171003817100381710038171198911003817100381710038171003817119862(119880)

le 119862 (120574 119904)1003817100381710038171003817119891

1003817100381710038171003817119867119904120574(119880)

(32)

(4) If 120572 gt 119904 and 1(1 + 120574(sdot)2) isin 119871

120572(119866

and) then 119867

119904

120574(119880) 997893rarr

119871120572lowast

(119880) where 120572lowast= 2120572(120572 minus 119904) Moreover there exists

a constant 119863(120574 119904) such that for each 119891 isin 119867119904

120574(119866) the

following inequality holds1003817100381710038171003817119891

1003817100381710038171003817119871120572lowast(119880)

le 119863 (120574 119904)1003817100381710038171003817119891

1003817100381710038171003817119867119904120574(119880)

(33)

We now prove the following compactness theorem indetail

Theorem 14 Let119866 be an LCA group and let119880 be a subset of119866of finite measure Assume that 1(1 + 1205742(sdot)) isin 119871

120572(119866

and) for some

120572 gt 119904 and that1003816100381610038161003816120585 (ℎ) minus 1

1003816100381610038161003816

(1 + 1205742 (120585))119904 997888rarr

ℎrarr1198900 119906119899119894119891119900119903119898119897119910 119908119894119905ℎ 119903119890119904119901119890119888119905 119905119900 120585 isin 119866

and

(34)

Then for all 119901 lt 120572lowast one has the compact embedding

119867119904

120574(119880) 997893rarr997893rarr 119871

119901(119880) (35)

The convergence concept used in (34) is explained in ourprevious paper [1] Let us mention that a condition similarto (34) appears in the characterization of precompact sets in1198712(119866) via Fourier transform see [30]

Proof We will need two lemmas which we proved in [1]

Lemma 15 Let 119865 isin 119867119904

120574(119866) and assume that

1003816100381610038161003816120585 (ℎ) minus 11003816100381610038161003816

(1 + 1205742 (120585))119904 997888rarr

ℎrarr1198900 119906119899119894119891119900119903119898119897119910 119908119894119905ℎ 119903119890119904119901119890119888119905 119905119900 120585 isin 119866

and

(36)

Then for every ℎ isin 119866

int119866

|119865 (119909ℎ) minus 119865 (119909)|2119889119909 le 119862 (ℎ) 119865

2

119867119904120574(119866)

(37)

where 119862(ℎ) rarrℎrarr119890

0

Journal of Function Spaces 5

Lemma 16 Let us denote by I the set of all symmetric unitneighbourhoods of group G partially ordered by inclusion Let(120601119881)119881isinI be a Dirac net and let 119865 isin 119867

119904

120574(119866) Then

int119866

1003816100381610038161003816119865 lowast 120601119881 (119909) minus 119865 (119909)1003816100381610038161003816

2119889119909 le 119865119867119904

120574(119866)sup

119910isin119881

119862 (119910) (38)

Nowwe can continuewith the proof ofTheorem 14 Let119891119899

be a bounded sequence in119867119904

120574(119880) that is 119891119899119867119904

120574(119880)

le 119862 Fromthe very definition there exist functions 119865119899 such that 119865119899 = 119891119899

in119880119865119899 isin 119867119904

120574(119866) and 119865119899119867119904

120574(119866)

le 119891119899119867119904120574(119880)

+1 for 119899 = 1 2 Now by Theorem 13 we get that the sequence 119891119899 is boundedin 119871

120572lowast

(119880) Hence there exists a subsequence 119891119899119896such that

119891119899119896 119891 in 119871

120572lowast

(119880) and there also exists a subsequence 119865119899119896

such that 119865119899119896 119865 in 119871

120572lowast

(119866) For simplicity we simply write119891119899 and 119865119899 for these subsequences From the basic propertiesof weak convergence we have that 119865 = 119891 in 119880 We will showthat for 119902 lt 120572

lowast we have

119891119899 997888rarr 119891 in 119871119902(119880) (39)

Let us set 119865(119881) = 119865 lowast 120601119881 From the triangle inequality we get

1003817100381710038171003817119891119899 minus 11989110038171003817100381710038171198712(119880)

le1003817100381710038171003817119891119899 minus 119865119899(119881)

10038171003817100381710038171198712(119880)

+1003817100381710038171003817119865119899(119881) minus 119865(119881)

10038171003817100381710038171198712(119880)+1003817100381710038171003817119865(119881) minus 119891

10038171003817100381710038171198712(119880)

le1003817100381710038171003817119865119899 minus 119865119899(119881)

10038171003817100381710038171198712(119866)+1003817100381710038171003817119865119899(119881) minus 119865(119881)

10038171003817100381710038171198712(119880)

+1003817100381710038171003817119865(119881) minus 119865

10038171003817100381710038171198712(119866)

(40)

From Lemma 15 we get

sup119899

int119866

10038161003816100381610038161003816119865119899(119881)

(119909) minus 119865119899 (119909)10038161003816100381610038161003816

2

119889119909 le sup119899

1003817100381710038171003817119865119899

1003817100381710038171003817

2

119867119904120574(119866)

sup119910isin119881

119862 (119910)

le (119862 + 1)2sup119910isin119881

119862 (119910)

(41)

Furthermore it can be easily shown that for each 120598 gt 0 thereexists 119881120598 such that for all 119881 isin I with 119881 sub 119881120598 we have

1003817100381710038171003817119865(119881) minus 11986510038171003817100381710038171198712(119866)

le120598

3 (42)

Hance we can choose 119881120598 such that

1003817100381710038171003817119891119899 minus 11989110038171003817100381710038171198712(119880)

le2

3120598 +

10038171003817100381710038171003817119865119899(119881120598)

minus 119865(119881120598)

100381710038171003817100381710038171198712(119880) (43)

We need to estimate 119865119899(119881120598)minus 119865(119881120598)

1198712(119880)

Since 119865119899 119865 in119871120572lowast

(119866) we get that 119865119899(119881120598)(119909) rarr

119899rarrinfin119865(119881120598)

(119909) Moreover wehave

10038161003816100381610038161003816119865119899(119881120598)

(119909) minus 119865(119881120598)(119909)

10038161003816100381610038161003816

2

=

10038161003816100381610038161003816100381610038161003816

int119866

(119865119899(119910) minus 119865(119910))120601119881120598(119910

minus1119909)119889119910

10038161003816100381610038161003816100381610038161003816

2

=

10038161003816100381610038161003816100381610038161003816

int119866

(119865119899(119910) minus 119865(119910))radic120601119881120598(119910minus1119909)radic120601119881120598

(119910minus1119909)119889119910

10038161003816100381610038161003816100381610038161003816

2

le int119866

120601119881120598(119910

minus1119909) 119889120583119866 (119910)

times int119866

1003816100381610038161003816119865119899 (119910) minus 119865 (119910)1003816100381610038161003816

2120601119881120598

(119910minus1119909) 119889119910

le sup119911isin119881120598

120601119881120598(119911) int

119866

1003816100381610038161003816119865119899 (119910) minus 119865 (119910)1003816100381610038161003816

2120601119881120598

(119910minus1119909) 119889119910

(44)

Hence since 119880 has a finite measure we get from Lebesgueconvergence theorem that

10038171003817100381710038171003817119865119899(119881120598)

minus 119865(119881120598)

100381710038171003817100381710038171198712(119880)le120598

3 (45)

We conclude that 119891119899 rarr 119891 in 1198712(119880) In order to finish the

proof it is enough to use the Vitali convergence theoremWeconclude that 119891119899 rarr 119891 in 119871

119902(119880) for 119902 lt 120572

lowast

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

Przemysław Gorkarsquos research is partially supported by theEuropean Union in the framework of European Social Fundthrough the Warsaw University of Technology DevelopmentProgram EnriqueG Reyesrsquo research is partially supported bythe Project FONDECYT no 1111042

References

[1] P Gorka and E G Reyes ldquoSobolev spaceson locally compactabelian groupsrdquo httparXivorgabs12083053 in press

[2] P Gorka T Kostrzewa and E G Reyes ldquoThe Rellich lemmaon compact abelian groups and equations of infinite orderrdquoInternational Journal of Geometric Methods in Modern Physicsvol 10 no 2 Article ID 1220030 11 pages 2013

[3] R A Adams Sobolev Spaces Academic Press New York NYUSA 1975

[4] M E Taylor Partial Differential Equations vol 23 SpringerNew York NY USA 1996

[5] T Aubin Some Nonlinear Problems in Riemannian GeometrySpringer Berlin Germany 1998

[6] E Hebey Sobolev Spaces on Riemannian Manifolds vol 1635Springer Berlin Germany 1996

[7] P Hajłasz ldquoSobolev spaces on an arbitrary metric spacerdquoPotential Analysis vol 5 no 4 pp 403ndash415 1996

[8] P Hajłasz and P Koskela ldquoSobolev met Poincarerdquo Memoirs ofthe American Mathematical Society vol 145 no 688 2000

[9] N Shanmugalingam ldquoNewtonian spaces an extension ofSobolev spaces to metric measure spacesrdquo Revista MatematicaIberoamericana vol 16 no 2 pp 243ndash279 2000

6 Journal of Function Spaces

[10] Y-C Kim ldquoA simple proof of the 119901-adic version of the Sobolevembedding theoremrdquo Korean Mathematical Society vol 25 no1 pp 27ndash36 2010

[11] J J Rodrıguez-Vega and W A Zuniga-Galindo ldquoElliptic pseu-dodifferential equations and Sobolev spaces over 119901-adic fieldsrdquoPacific Journal ofMathematics vol 246 no 2 pp 407ndash420 2010

[12] H Bahouri C Fermanian-Kammerer and I Gallagher ldquoAnal-yse de lrsquoespace des phases et calcul pseudo-differential sur legroupe de Heisenbergrdquo Comptes Rendus Mathematique vol347 no 17-18 pp 1021ndash1024 2009

[13] P Gorka H Prado and E G Reyes ldquoNonlinear equations withinfinitely many derivativesrdquo Complex Analysis and OperatorTheory vol 5 no 1 pp 313ndash323 2011

[14] P Gorka H Prado and E G Reyes ldquoGeneralized euclideanbosonic string equationsrdquo in Operator Theory Advances andApplications vol 224 pp 147ndash169 Springer Basel Switzerland2012

[15] P Gorka H Prado and E G Reyes ldquoOn a general class ofnonlocal equationsrdquo Annales Henri Poincare vol 14 no 4 pp947ndash966 2013

[16] P Gorka H Prado and E G Reyes ldquoFunctional calculusvia Laplace transform and equations with infinitely manyderivativesrdquo Journal of Mathematical Physics vol 51 no 10Article ID 103512 2010

[17] P Gorka H Prado and E G Reyes ldquoThe initial value problemfor ordinary differential equations with infinitely many deriva-tivesrdquo Classical and Quantum Gravity vol 29 no 6 Article ID065017 2012

[18] N Barnaby and N Kamran ldquoDynamics with infinitely manyderivatives the initial value problemrdquo Journal of High EnergyPhysics no 2 40 pages 2008

[19] G Calcagni M Montobbio and G Nardelli ldquoLocalization ofnonlocal theoriesrdquo Physics Letters B vol 662 no 3 pp 285ndash2892008

[20] V S Vladimirov ldquoOn the equation of a 119901-adic open string for ascalar tachyon fieldrdquo Izvestiya vol 69 no 3 pp 487ndash512 2005

[21] E Witten ldquoNoncommutative geometry and string field theoryrdquoNuclear Physics B vol 268 no 2 pp 253ndash294 1986

[22] D Zhou and C J C Burges ldquoHigh-order regularization ongraphsrdquo in Proceedings of the 6th International Workshop onMining and Learning with Graphs Helsinki Finland 2008httpwwwcishutfiMLG08paperspostersessionpaper39pdf

[23] H Triebel ldquoWavelets in function spaces on Lipschitz domainsrdquoMathematische Nachrichten vol 280 no 9-10 pp 1205ndash12182007

[24] E Hewitt andK A RossAbstract Harmonic Analysis Vol I DieGrundlehren der mathematischen Wissenschaften SpringerBerlin Germany 1963

[25] E Hewitt and K A Ross Abstract Harmonic Analysis VolII Die Grundlehren der mathematischen WissenschaftenSpringer Berlin Germany 1970

[26] H G Feichtinger and T Werther ldquoRobustness of regularsampling in Sobolev algebrasrdquo in Sampling Wavelets andTomography J Benedetto and A I Zayed Eds pp 83ndash113Birkhauser Boston Boston Mass USA 2004

[27] E Popescu ldquoNonlocal Dirichlet forms generated by pseudod-ifferential operators on compact abelian groupsrdquo Proceedingsin Applied Mathematics and Mechanics vol 7 pp 2160001ndash2160002 2007

[28] A Weil LrsquoIntegration dans les Groupes Topologiques et ses Appli-cations Hermann et Cie Paris France 1940

[29] H Triebel ldquoFunction spaces in Lipschitz domains and onLipschitz manifolds Characteristic functions as pointwise mul-tipliersrdquo Revista Matematica Complutense vol 15 no 2 pp475ndash524 2002

[30] P Gorka ldquoPego theorem on locally compact abelian groupsrdquoJournal of Algebra and Its Applications vol 13 no 4 Article ID1350143 2014

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

4 Journal of Function Spaces

Proof Using Lemma 8 for the function 1(1 + 1205742(sdot))

120572 we seethat 120574 satisfies the assumptions ofTheorem 6 so for 119904 lt 120572 weget that

119867119904

120574(119866) 997893rarr997893rarr 119867

120572

120574(119866) (24)

Moreover thanks to the first part of Theorem 5 we have

119867120572

120574(119866) 997893rarr 119862 (119866) (25)

and the proof follows

Theorem11 If119866 is compact 1(1+1205742(sdot)) isin 119871120572(119866

and) and 119904 lt 120572

then

119867119904

120574(119866) 997893rarr997893rarr 119871

119901(119866) (26)

for all 119901 lt 120572lowast= 2120572(120572 minus 119904)

Proof Let us take 120575 = 120572 minus 2120572119901 then 120575 lt 119904 Thus byTheorem 6 we have the compact embedding

119867119904

120574(119866) 997893rarr997893rarr 119867

120575

120574(119866) (27)

Next using the second part of Theorem 5 we have thecontinuous embedding

119867120575

120574(119866) 997893rarr 119871

120572lowast

120575 (119866) (28)

where 120572lowast

120575= 2120572(120572 minus 120575) = 119901

4 Sobolev Spaces on Subsets of LCA Groups

In this section we deal with Sobolev spaces defined onsubsets of locally compact abelian groups As mentioned inSection 1 we are motivated by analogous studies of functionspaces on domains of R119899 (see eg [29]) by the fact thatinteresting applications exist [23] and by the possibility ofusing them as tools for the study of differential equations onsubsets of LCA groupsWe start with the following definition

Definition 12 Let119880 be a subset of a LCA group 119866 We definethe Sobolev space 119867

119904

120574(119880) sub 119871

2(119880) as the space of all 119891 isin

1198712(119880) such that there exists 119865 isin 119867

119904

120574(119866) with 119865|119880 = 119891 and

we equip it with the norm

10038171003817100381710038171198911003817100381710038171003817119867119904120574(119880)

= inf 119865119867119904120574(119866) 119865 isin 119867

119904

120574(119866) and 119865|119880 = 119891

(29)

An analogous definition (of spaces119861119904

119901119902on domains ofR119899)

appears in [29] see his Definition 23 It can be easily shownthat 119867119904

120574(119880) is a Banach space We will say that it is a local

Sobolev spaceUsing appropriate embeddings for119867119904

120574(119866) and the defini-

tion of119867119904

120574(119880) we can prove the following

Theorem 13 Let 119866 be a locally compact abelian group and let119880 sub 119866 Then we have

(1) The continuous inclusion 119867119904

120574(119880) 997893rarr 119871

2(119880) holds

Moreover for each 119891 isin 119867119904

120574(119880) the following inequality

holds1003817100381710038171003817119891

10038171003817100381710038171198712(119880)le1003817100381710038171003817119891

1003817100381710038171003817119867119904120574(119880)

(30)

(2) If 119904 gt 120590 then119867119904

120574(119866) 997893rarr 119867

120590

120574(119880) Moreover the follow-

ing inequality holds1003817100381710038171003817119891

1003817100381710038171003817119867120590120574(119880)

le1003817100381710038171003817119891

1003817100381710038171003817119867119904120574(119880)

(31)

(3) If 1(1 + 120574(sdot)2) isin 119871

119904(119866

and) then 119867

119904

120574(119880) 997893rarr 119862(119880)

Moreover there exists a constant 119862(120574 119904) such that foreach 119891 isin 119867

119904

120574(119880) the following inequality holds

10038171003817100381710038171198911003817100381710038171003817119862(119880)

le 119862 (120574 119904)1003817100381710038171003817119891

1003817100381710038171003817119867119904120574(119880)

(32)

(4) If 120572 gt 119904 and 1(1 + 120574(sdot)2) isin 119871

120572(119866

and) then 119867

119904

120574(119880) 997893rarr

119871120572lowast

(119880) where 120572lowast= 2120572(120572 minus 119904) Moreover there exists

a constant 119863(120574 119904) such that for each 119891 isin 119867119904

120574(119866) the

following inequality holds1003817100381710038171003817119891

1003817100381710038171003817119871120572lowast(119880)

le 119863 (120574 119904)1003817100381710038171003817119891

1003817100381710038171003817119867119904120574(119880)

(33)

We now prove the following compactness theorem indetail

Theorem 14 Let119866 be an LCA group and let119880 be a subset of119866of finite measure Assume that 1(1 + 1205742(sdot)) isin 119871

120572(119866

and) for some

120572 gt 119904 and that1003816100381610038161003816120585 (ℎ) minus 1

1003816100381610038161003816

(1 + 1205742 (120585))119904 997888rarr

ℎrarr1198900 119906119899119894119891119900119903119898119897119910 119908119894119905ℎ 119903119890119904119901119890119888119905 119905119900 120585 isin 119866

and

(34)

Then for all 119901 lt 120572lowast one has the compact embedding

119867119904

120574(119880) 997893rarr997893rarr 119871

119901(119880) (35)

The convergence concept used in (34) is explained in ourprevious paper [1] Let us mention that a condition similarto (34) appears in the characterization of precompact sets in1198712(119866) via Fourier transform see [30]

Proof We will need two lemmas which we proved in [1]

Lemma 15 Let 119865 isin 119867119904

120574(119866) and assume that

1003816100381610038161003816120585 (ℎ) minus 11003816100381610038161003816

(1 + 1205742 (120585))119904 997888rarr

ℎrarr1198900 119906119899119894119891119900119903119898119897119910 119908119894119905ℎ 119903119890119904119901119890119888119905 119905119900 120585 isin 119866

and

(36)

Then for every ℎ isin 119866

int119866

|119865 (119909ℎ) minus 119865 (119909)|2119889119909 le 119862 (ℎ) 119865

2

119867119904120574(119866)

(37)

where 119862(ℎ) rarrℎrarr119890

0

Journal of Function Spaces 5

Lemma 16 Let us denote by I the set of all symmetric unitneighbourhoods of group G partially ordered by inclusion Let(120601119881)119881isinI be a Dirac net and let 119865 isin 119867

119904

120574(119866) Then

int119866

1003816100381610038161003816119865 lowast 120601119881 (119909) minus 119865 (119909)1003816100381610038161003816

2119889119909 le 119865119867119904

120574(119866)sup

119910isin119881

119862 (119910) (38)

Nowwe can continuewith the proof ofTheorem 14 Let119891119899

be a bounded sequence in119867119904

120574(119880) that is 119891119899119867119904

120574(119880)

le 119862 Fromthe very definition there exist functions 119865119899 such that 119865119899 = 119891119899

in119880119865119899 isin 119867119904

120574(119866) and 119865119899119867119904

120574(119866)

le 119891119899119867119904120574(119880)

+1 for 119899 = 1 2 Now by Theorem 13 we get that the sequence 119891119899 is boundedin 119871

120572lowast

(119880) Hence there exists a subsequence 119891119899119896such that

119891119899119896 119891 in 119871

120572lowast

(119880) and there also exists a subsequence 119865119899119896

such that 119865119899119896 119865 in 119871

120572lowast

(119866) For simplicity we simply write119891119899 and 119865119899 for these subsequences From the basic propertiesof weak convergence we have that 119865 = 119891 in 119880 We will showthat for 119902 lt 120572

lowast we have

119891119899 997888rarr 119891 in 119871119902(119880) (39)

Let us set 119865(119881) = 119865 lowast 120601119881 From the triangle inequality we get

1003817100381710038171003817119891119899 minus 11989110038171003817100381710038171198712(119880)

le1003817100381710038171003817119891119899 minus 119865119899(119881)

10038171003817100381710038171198712(119880)

+1003817100381710038171003817119865119899(119881) minus 119865(119881)

10038171003817100381710038171198712(119880)+1003817100381710038171003817119865(119881) minus 119891

10038171003817100381710038171198712(119880)

le1003817100381710038171003817119865119899 minus 119865119899(119881)

10038171003817100381710038171198712(119866)+1003817100381710038171003817119865119899(119881) minus 119865(119881)

10038171003817100381710038171198712(119880)

+1003817100381710038171003817119865(119881) minus 119865

10038171003817100381710038171198712(119866)

(40)

From Lemma 15 we get

sup119899

int119866

10038161003816100381610038161003816119865119899(119881)

(119909) minus 119865119899 (119909)10038161003816100381610038161003816

2

119889119909 le sup119899

1003817100381710038171003817119865119899

1003817100381710038171003817

2

119867119904120574(119866)

sup119910isin119881

119862 (119910)

le (119862 + 1)2sup119910isin119881

119862 (119910)

(41)

Furthermore it can be easily shown that for each 120598 gt 0 thereexists 119881120598 such that for all 119881 isin I with 119881 sub 119881120598 we have

1003817100381710038171003817119865(119881) minus 11986510038171003817100381710038171198712(119866)

le120598

3 (42)

Hance we can choose 119881120598 such that

1003817100381710038171003817119891119899 minus 11989110038171003817100381710038171198712(119880)

le2

3120598 +

10038171003817100381710038171003817119865119899(119881120598)

minus 119865(119881120598)

100381710038171003817100381710038171198712(119880) (43)

We need to estimate 119865119899(119881120598)minus 119865(119881120598)

1198712(119880)

Since 119865119899 119865 in119871120572lowast

(119866) we get that 119865119899(119881120598)(119909) rarr

119899rarrinfin119865(119881120598)

(119909) Moreover wehave

10038161003816100381610038161003816119865119899(119881120598)

(119909) minus 119865(119881120598)(119909)

10038161003816100381610038161003816

2

=

10038161003816100381610038161003816100381610038161003816

int119866

(119865119899(119910) minus 119865(119910))120601119881120598(119910

minus1119909)119889119910

10038161003816100381610038161003816100381610038161003816

2

=

10038161003816100381610038161003816100381610038161003816

int119866

(119865119899(119910) minus 119865(119910))radic120601119881120598(119910minus1119909)radic120601119881120598

(119910minus1119909)119889119910

10038161003816100381610038161003816100381610038161003816

2

le int119866

120601119881120598(119910

minus1119909) 119889120583119866 (119910)

times int119866

1003816100381610038161003816119865119899 (119910) minus 119865 (119910)1003816100381610038161003816

2120601119881120598

(119910minus1119909) 119889119910

le sup119911isin119881120598

120601119881120598(119911) int

119866

1003816100381610038161003816119865119899 (119910) minus 119865 (119910)1003816100381610038161003816

2120601119881120598

(119910minus1119909) 119889119910

(44)

Hence since 119880 has a finite measure we get from Lebesgueconvergence theorem that

10038171003817100381710038171003817119865119899(119881120598)

minus 119865(119881120598)

100381710038171003817100381710038171198712(119880)le120598

3 (45)

We conclude that 119891119899 rarr 119891 in 1198712(119880) In order to finish the

proof it is enough to use the Vitali convergence theoremWeconclude that 119891119899 rarr 119891 in 119871

119902(119880) for 119902 lt 120572

lowast

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

Przemysław Gorkarsquos research is partially supported by theEuropean Union in the framework of European Social Fundthrough the Warsaw University of Technology DevelopmentProgram EnriqueG Reyesrsquo research is partially supported bythe Project FONDECYT no 1111042

References

[1] P Gorka and E G Reyes ldquoSobolev spaceson locally compactabelian groupsrdquo httparXivorgabs12083053 in press

[2] P Gorka T Kostrzewa and E G Reyes ldquoThe Rellich lemmaon compact abelian groups and equations of infinite orderrdquoInternational Journal of Geometric Methods in Modern Physicsvol 10 no 2 Article ID 1220030 11 pages 2013

[3] R A Adams Sobolev Spaces Academic Press New York NYUSA 1975

[4] M E Taylor Partial Differential Equations vol 23 SpringerNew York NY USA 1996

[5] T Aubin Some Nonlinear Problems in Riemannian GeometrySpringer Berlin Germany 1998

[6] E Hebey Sobolev Spaces on Riemannian Manifolds vol 1635Springer Berlin Germany 1996

[7] P Hajłasz ldquoSobolev spaces on an arbitrary metric spacerdquoPotential Analysis vol 5 no 4 pp 403ndash415 1996

[8] P Hajłasz and P Koskela ldquoSobolev met Poincarerdquo Memoirs ofthe American Mathematical Society vol 145 no 688 2000

[9] N Shanmugalingam ldquoNewtonian spaces an extension ofSobolev spaces to metric measure spacesrdquo Revista MatematicaIberoamericana vol 16 no 2 pp 243ndash279 2000

6 Journal of Function Spaces

[10] Y-C Kim ldquoA simple proof of the 119901-adic version of the Sobolevembedding theoremrdquo Korean Mathematical Society vol 25 no1 pp 27ndash36 2010

[11] J J Rodrıguez-Vega and W A Zuniga-Galindo ldquoElliptic pseu-dodifferential equations and Sobolev spaces over 119901-adic fieldsrdquoPacific Journal ofMathematics vol 246 no 2 pp 407ndash420 2010

[12] H Bahouri C Fermanian-Kammerer and I Gallagher ldquoAnal-yse de lrsquoespace des phases et calcul pseudo-differential sur legroupe de Heisenbergrdquo Comptes Rendus Mathematique vol347 no 17-18 pp 1021ndash1024 2009

[13] P Gorka H Prado and E G Reyes ldquoNonlinear equations withinfinitely many derivativesrdquo Complex Analysis and OperatorTheory vol 5 no 1 pp 313ndash323 2011

[14] P Gorka H Prado and E G Reyes ldquoGeneralized euclideanbosonic string equationsrdquo in Operator Theory Advances andApplications vol 224 pp 147ndash169 Springer Basel Switzerland2012

[15] P Gorka H Prado and E G Reyes ldquoOn a general class ofnonlocal equationsrdquo Annales Henri Poincare vol 14 no 4 pp947ndash966 2013

[16] P Gorka H Prado and E G Reyes ldquoFunctional calculusvia Laplace transform and equations with infinitely manyderivativesrdquo Journal of Mathematical Physics vol 51 no 10Article ID 103512 2010

[17] P Gorka H Prado and E G Reyes ldquoThe initial value problemfor ordinary differential equations with infinitely many deriva-tivesrdquo Classical and Quantum Gravity vol 29 no 6 Article ID065017 2012

[18] N Barnaby and N Kamran ldquoDynamics with infinitely manyderivatives the initial value problemrdquo Journal of High EnergyPhysics no 2 40 pages 2008

[19] G Calcagni M Montobbio and G Nardelli ldquoLocalization ofnonlocal theoriesrdquo Physics Letters B vol 662 no 3 pp 285ndash2892008

[20] V S Vladimirov ldquoOn the equation of a 119901-adic open string for ascalar tachyon fieldrdquo Izvestiya vol 69 no 3 pp 487ndash512 2005

[21] E Witten ldquoNoncommutative geometry and string field theoryrdquoNuclear Physics B vol 268 no 2 pp 253ndash294 1986

[22] D Zhou and C J C Burges ldquoHigh-order regularization ongraphsrdquo in Proceedings of the 6th International Workshop onMining and Learning with Graphs Helsinki Finland 2008httpwwwcishutfiMLG08paperspostersessionpaper39pdf

[23] H Triebel ldquoWavelets in function spaces on Lipschitz domainsrdquoMathematische Nachrichten vol 280 no 9-10 pp 1205ndash12182007

[24] E Hewitt andK A RossAbstract Harmonic Analysis Vol I DieGrundlehren der mathematischen Wissenschaften SpringerBerlin Germany 1963

[25] E Hewitt and K A Ross Abstract Harmonic Analysis VolII Die Grundlehren der mathematischen WissenschaftenSpringer Berlin Germany 1970

[26] H G Feichtinger and T Werther ldquoRobustness of regularsampling in Sobolev algebrasrdquo in Sampling Wavelets andTomography J Benedetto and A I Zayed Eds pp 83ndash113Birkhauser Boston Boston Mass USA 2004

[27] E Popescu ldquoNonlocal Dirichlet forms generated by pseudod-ifferential operators on compact abelian groupsrdquo Proceedingsin Applied Mathematics and Mechanics vol 7 pp 2160001ndash2160002 2007

[28] A Weil LrsquoIntegration dans les Groupes Topologiques et ses Appli-cations Hermann et Cie Paris France 1940

[29] H Triebel ldquoFunction spaces in Lipschitz domains and onLipschitz manifolds Characteristic functions as pointwise mul-tipliersrdquo Revista Matematica Complutense vol 15 no 2 pp475ndash524 2002

[30] P Gorka ldquoPego theorem on locally compact abelian groupsrdquoJournal of Algebra and Its Applications vol 13 no 4 Article ID1350143 2014

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Journal of Function Spaces 5

Lemma 16 Let us denote by I the set of all symmetric unitneighbourhoods of group G partially ordered by inclusion Let(120601119881)119881isinI be a Dirac net and let 119865 isin 119867

119904

120574(119866) Then

int119866

1003816100381610038161003816119865 lowast 120601119881 (119909) minus 119865 (119909)1003816100381610038161003816

2119889119909 le 119865119867119904

120574(119866)sup

119910isin119881

119862 (119910) (38)

Nowwe can continuewith the proof ofTheorem 14 Let119891119899

be a bounded sequence in119867119904

120574(119880) that is 119891119899119867119904

120574(119880)

le 119862 Fromthe very definition there exist functions 119865119899 such that 119865119899 = 119891119899

in119880119865119899 isin 119867119904

120574(119866) and 119865119899119867119904

120574(119866)

le 119891119899119867119904120574(119880)

+1 for 119899 = 1 2 Now by Theorem 13 we get that the sequence 119891119899 is boundedin 119871

120572lowast

(119880) Hence there exists a subsequence 119891119899119896such that

119891119899119896 119891 in 119871

120572lowast

(119880) and there also exists a subsequence 119865119899119896

such that 119865119899119896 119865 in 119871

120572lowast

(119866) For simplicity we simply write119891119899 and 119865119899 for these subsequences From the basic propertiesof weak convergence we have that 119865 = 119891 in 119880 We will showthat for 119902 lt 120572

lowast we have

119891119899 997888rarr 119891 in 119871119902(119880) (39)

Let us set 119865(119881) = 119865 lowast 120601119881 From the triangle inequality we get

1003817100381710038171003817119891119899 minus 11989110038171003817100381710038171198712(119880)

le1003817100381710038171003817119891119899 minus 119865119899(119881)

10038171003817100381710038171198712(119880)

+1003817100381710038171003817119865119899(119881) minus 119865(119881)

10038171003817100381710038171198712(119880)+1003817100381710038171003817119865(119881) minus 119891

10038171003817100381710038171198712(119880)

le1003817100381710038171003817119865119899 minus 119865119899(119881)

10038171003817100381710038171198712(119866)+1003817100381710038171003817119865119899(119881) minus 119865(119881)

10038171003817100381710038171198712(119880)

+1003817100381710038171003817119865(119881) minus 119865

10038171003817100381710038171198712(119866)

(40)

From Lemma 15 we get

sup119899

int119866

10038161003816100381610038161003816119865119899(119881)

(119909) minus 119865119899 (119909)10038161003816100381610038161003816

2

119889119909 le sup119899

1003817100381710038171003817119865119899

1003817100381710038171003817

2

119867119904120574(119866)

sup119910isin119881

119862 (119910)

le (119862 + 1)2sup119910isin119881

119862 (119910)

(41)

Furthermore it can be easily shown that for each 120598 gt 0 thereexists 119881120598 such that for all 119881 isin I with 119881 sub 119881120598 we have

1003817100381710038171003817119865(119881) minus 11986510038171003817100381710038171198712(119866)

le120598

3 (42)

Hance we can choose 119881120598 such that

1003817100381710038171003817119891119899 minus 11989110038171003817100381710038171198712(119880)

le2

3120598 +

10038171003817100381710038171003817119865119899(119881120598)

minus 119865(119881120598)

100381710038171003817100381710038171198712(119880) (43)

We need to estimate 119865119899(119881120598)minus 119865(119881120598)

1198712(119880)

Since 119865119899 119865 in119871120572lowast

(119866) we get that 119865119899(119881120598)(119909) rarr

119899rarrinfin119865(119881120598)

(119909) Moreover wehave

10038161003816100381610038161003816119865119899(119881120598)

(119909) minus 119865(119881120598)(119909)

10038161003816100381610038161003816

2

=

10038161003816100381610038161003816100381610038161003816

int119866

(119865119899(119910) minus 119865(119910))120601119881120598(119910

minus1119909)119889119910

10038161003816100381610038161003816100381610038161003816

2

=

10038161003816100381610038161003816100381610038161003816

int119866

(119865119899(119910) minus 119865(119910))radic120601119881120598(119910minus1119909)radic120601119881120598

(119910minus1119909)119889119910

10038161003816100381610038161003816100381610038161003816

2

le int119866

120601119881120598(119910

minus1119909) 119889120583119866 (119910)

times int119866

1003816100381610038161003816119865119899 (119910) minus 119865 (119910)1003816100381610038161003816

2120601119881120598

(119910minus1119909) 119889119910

le sup119911isin119881120598

120601119881120598(119911) int

119866

1003816100381610038161003816119865119899 (119910) minus 119865 (119910)1003816100381610038161003816

2120601119881120598

(119910minus1119909) 119889119910

(44)

Hence since 119880 has a finite measure we get from Lebesgueconvergence theorem that

10038171003817100381710038171003817119865119899(119881120598)

minus 119865(119881120598)

100381710038171003817100381710038171198712(119880)le120598

3 (45)

We conclude that 119891119899 rarr 119891 in 1198712(119880) In order to finish the

proof it is enough to use the Vitali convergence theoremWeconclude that 119891119899 rarr 119891 in 119871

119902(119880) for 119902 lt 120572

lowast

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

Przemysław Gorkarsquos research is partially supported by theEuropean Union in the framework of European Social Fundthrough the Warsaw University of Technology DevelopmentProgram EnriqueG Reyesrsquo research is partially supported bythe Project FONDECYT no 1111042

References

[1] P Gorka and E G Reyes ldquoSobolev spaceson locally compactabelian groupsrdquo httparXivorgabs12083053 in press

[2] P Gorka T Kostrzewa and E G Reyes ldquoThe Rellich lemmaon compact abelian groups and equations of infinite orderrdquoInternational Journal of Geometric Methods in Modern Physicsvol 10 no 2 Article ID 1220030 11 pages 2013

[3] R A Adams Sobolev Spaces Academic Press New York NYUSA 1975

[4] M E Taylor Partial Differential Equations vol 23 SpringerNew York NY USA 1996

[5] T Aubin Some Nonlinear Problems in Riemannian GeometrySpringer Berlin Germany 1998

[6] E Hebey Sobolev Spaces on Riemannian Manifolds vol 1635Springer Berlin Germany 1996

[7] P Hajłasz ldquoSobolev spaces on an arbitrary metric spacerdquoPotential Analysis vol 5 no 4 pp 403ndash415 1996

[8] P Hajłasz and P Koskela ldquoSobolev met Poincarerdquo Memoirs ofthe American Mathematical Society vol 145 no 688 2000

[9] N Shanmugalingam ldquoNewtonian spaces an extension ofSobolev spaces to metric measure spacesrdquo Revista MatematicaIberoamericana vol 16 no 2 pp 243ndash279 2000

6 Journal of Function Spaces

[10] Y-C Kim ldquoA simple proof of the 119901-adic version of the Sobolevembedding theoremrdquo Korean Mathematical Society vol 25 no1 pp 27ndash36 2010

[11] J J Rodrıguez-Vega and W A Zuniga-Galindo ldquoElliptic pseu-dodifferential equations and Sobolev spaces over 119901-adic fieldsrdquoPacific Journal ofMathematics vol 246 no 2 pp 407ndash420 2010

[12] H Bahouri C Fermanian-Kammerer and I Gallagher ldquoAnal-yse de lrsquoespace des phases et calcul pseudo-differential sur legroupe de Heisenbergrdquo Comptes Rendus Mathematique vol347 no 17-18 pp 1021ndash1024 2009

[13] P Gorka H Prado and E G Reyes ldquoNonlinear equations withinfinitely many derivativesrdquo Complex Analysis and OperatorTheory vol 5 no 1 pp 313ndash323 2011

[14] P Gorka H Prado and E G Reyes ldquoGeneralized euclideanbosonic string equationsrdquo in Operator Theory Advances andApplications vol 224 pp 147ndash169 Springer Basel Switzerland2012

[15] P Gorka H Prado and E G Reyes ldquoOn a general class ofnonlocal equationsrdquo Annales Henri Poincare vol 14 no 4 pp947ndash966 2013

[16] P Gorka H Prado and E G Reyes ldquoFunctional calculusvia Laplace transform and equations with infinitely manyderivativesrdquo Journal of Mathematical Physics vol 51 no 10Article ID 103512 2010

[17] P Gorka H Prado and E G Reyes ldquoThe initial value problemfor ordinary differential equations with infinitely many deriva-tivesrdquo Classical and Quantum Gravity vol 29 no 6 Article ID065017 2012

[18] N Barnaby and N Kamran ldquoDynamics with infinitely manyderivatives the initial value problemrdquo Journal of High EnergyPhysics no 2 40 pages 2008

[19] G Calcagni M Montobbio and G Nardelli ldquoLocalization ofnonlocal theoriesrdquo Physics Letters B vol 662 no 3 pp 285ndash2892008

[20] V S Vladimirov ldquoOn the equation of a 119901-adic open string for ascalar tachyon fieldrdquo Izvestiya vol 69 no 3 pp 487ndash512 2005

[21] E Witten ldquoNoncommutative geometry and string field theoryrdquoNuclear Physics B vol 268 no 2 pp 253ndash294 1986

[22] D Zhou and C J C Burges ldquoHigh-order regularization ongraphsrdquo in Proceedings of the 6th International Workshop onMining and Learning with Graphs Helsinki Finland 2008httpwwwcishutfiMLG08paperspostersessionpaper39pdf

[23] H Triebel ldquoWavelets in function spaces on Lipschitz domainsrdquoMathematische Nachrichten vol 280 no 9-10 pp 1205ndash12182007

[24] E Hewitt andK A RossAbstract Harmonic Analysis Vol I DieGrundlehren der mathematischen Wissenschaften SpringerBerlin Germany 1963

[25] E Hewitt and K A Ross Abstract Harmonic Analysis VolII Die Grundlehren der mathematischen WissenschaftenSpringer Berlin Germany 1970

[26] H G Feichtinger and T Werther ldquoRobustness of regularsampling in Sobolev algebrasrdquo in Sampling Wavelets andTomography J Benedetto and A I Zayed Eds pp 83ndash113Birkhauser Boston Boston Mass USA 2004

[27] E Popescu ldquoNonlocal Dirichlet forms generated by pseudod-ifferential operators on compact abelian groupsrdquo Proceedingsin Applied Mathematics and Mechanics vol 7 pp 2160001ndash2160002 2007

[28] A Weil LrsquoIntegration dans les Groupes Topologiques et ses Appli-cations Hermann et Cie Paris France 1940

[29] H Triebel ldquoFunction spaces in Lipschitz domains and onLipschitz manifolds Characteristic functions as pointwise mul-tipliersrdquo Revista Matematica Complutense vol 15 no 2 pp475ndash524 2002

[30] P Gorka ldquoPego theorem on locally compact abelian groupsrdquoJournal of Algebra and Its Applications vol 13 no 4 Article ID1350143 2014

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

6 Journal of Function Spaces

[10] Y-C Kim ldquoA simple proof of the 119901-adic version of the Sobolevembedding theoremrdquo Korean Mathematical Society vol 25 no1 pp 27ndash36 2010

[11] J J Rodrıguez-Vega and W A Zuniga-Galindo ldquoElliptic pseu-dodifferential equations and Sobolev spaces over 119901-adic fieldsrdquoPacific Journal ofMathematics vol 246 no 2 pp 407ndash420 2010

[12] H Bahouri C Fermanian-Kammerer and I Gallagher ldquoAnal-yse de lrsquoespace des phases et calcul pseudo-differential sur legroupe de Heisenbergrdquo Comptes Rendus Mathematique vol347 no 17-18 pp 1021ndash1024 2009

[13] P Gorka H Prado and E G Reyes ldquoNonlinear equations withinfinitely many derivativesrdquo Complex Analysis and OperatorTheory vol 5 no 1 pp 313ndash323 2011

[14] P Gorka H Prado and E G Reyes ldquoGeneralized euclideanbosonic string equationsrdquo in Operator Theory Advances andApplications vol 224 pp 147ndash169 Springer Basel Switzerland2012

[15] P Gorka H Prado and E G Reyes ldquoOn a general class ofnonlocal equationsrdquo Annales Henri Poincare vol 14 no 4 pp947ndash966 2013

[16] P Gorka H Prado and E G Reyes ldquoFunctional calculusvia Laplace transform and equations with infinitely manyderivativesrdquo Journal of Mathematical Physics vol 51 no 10Article ID 103512 2010

[17] P Gorka H Prado and E G Reyes ldquoThe initial value problemfor ordinary differential equations with infinitely many deriva-tivesrdquo Classical and Quantum Gravity vol 29 no 6 Article ID065017 2012

[18] N Barnaby and N Kamran ldquoDynamics with infinitely manyderivatives the initial value problemrdquo Journal of High EnergyPhysics no 2 40 pages 2008

[19] G Calcagni M Montobbio and G Nardelli ldquoLocalization ofnonlocal theoriesrdquo Physics Letters B vol 662 no 3 pp 285ndash2892008

[20] V S Vladimirov ldquoOn the equation of a 119901-adic open string for ascalar tachyon fieldrdquo Izvestiya vol 69 no 3 pp 487ndash512 2005

[21] E Witten ldquoNoncommutative geometry and string field theoryrdquoNuclear Physics B vol 268 no 2 pp 253ndash294 1986

[22] D Zhou and C J C Burges ldquoHigh-order regularization ongraphsrdquo in Proceedings of the 6th International Workshop onMining and Learning with Graphs Helsinki Finland 2008httpwwwcishutfiMLG08paperspostersessionpaper39pdf

[23] H Triebel ldquoWavelets in function spaces on Lipschitz domainsrdquoMathematische Nachrichten vol 280 no 9-10 pp 1205ndash12182007

[24] E Hewitt andK A RossAbstract Harmonic Analysis Vol I DieGrundlehren der mathematischen Wissenschaften SpringerBerlin Germany 1963

[25] E Hewitt and K A Ross Abstract Harmonic Analysis VolII Die Grundlehren der mathematischen WissenschaftenSpringer Berlin Germany 1970

[26] H G Feichtinger and T Werther ldquoRobustness of regularsampling in Sobolev algebrasrdquo in Sampling Wavelets andTomography J Benedetto and A I Zayed Eds pp 83ndash113Birkhauser Boston Boston Mass USA 2004

[27] E Popescu ldquoNonlocal Dirichlet forms generated by pseudod-ifferential operators on compact abelian groupsrdquo Proceedingsin Applied Mathematics and Mechanics vol 7 pp 2160001ndash2160002 2007

[28] A Weil LrsquoIntegration dans les Groupes Topologiques et ses Appli-cations Hermann et Cie Paris France 1940

[29] H Triebel ldquoFunction spaces in Lipschitz domains and onLipschitz manifolds Characteristic functions as pointwise mul-tipliersrdquo Revista Matematica Complutense vol 15 no 2 pp475ndash524 2002

[30] P Gorka ldquoPego theorem on locally compact abelian groupsrdquoJournal of Algebra and Its Applications vol 13 no 4 Article ID1350143 2014

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of