· J. Aust. Math. Soc. 80 (2006), 229–262 SOME NEW BESOV AND TRIEBEL-LIZORKIN SPACES ASSOCIATED WITH PARA-ACCRETIVE FUNCTIONS ON SPACES OF HOMOGENEOUS TYPE DONGGUO DENG and DACHUN

J. Aust. Math. Soc.80 (2006), 229–262

SOME NEW BESOV AND TRIEBEL-LIZORKIN SPACESASSOCIATED WITH PARA-ACCRETIVE FUNCTIONS ON SPACES

OF HOMOGENEOUS TYPE

DONGGUO DENG and DACHUN YANG

(Received 8 April 2004; revised 27 December 2004)

Communicated by T. Dooley

Abstract

Let .X; ²; ¼/d;� be a space of homogeneous type withd > 0 and� ∈ .0;1], b be a para-accretivefunction,ž ∈ .0; �], |s| < ž, anda0 ∈ .0;1/ be some constant depending ond, ž ands. The authorsintroduce the Besov spacebBs

pq.X/ with a0 < p ≤ ∞ and 0< q ≤ ∞, and the Triebel-Lizorkin spacebFs

pq.X/ with a0 < p < ∞ anda0 < q ≤ ∞ by first establishing a Plancherel-Polya-type inequality.Moreover, the authors establish the frame and the Littlewood-Paley function characterizations of thesespaces. Furthermore, the authors introduce the new Besov spaceb−1 Bs

pq.X/ and the Triebel-Lizorkinspaceb−1 Fs

pq.X/. The relations among these spaces and the known Hardy-type spaces are presented.As applications, the authors also establish some real interpolation theorems, embedding theorems,T btheorems, and the lifting property by introducing some new Riesz operators of these spaces.

2000Mathematics subject classification: primary primary 42B35; secondary 46E35, 42B25, 43A99,47B06, 42B20, 47A30, 47B38.Keywords and phrases: space of homogeneous type, para-accretive function, Plancherel-Polya inequality,

Besov space, Triebel-Lizorkin space, Calderon reproducing formula, interpolation, embedding theorem,T b theorem, Riesz potential, lifting property.

1. Introduction

It is well known that the remarkableT1 theorem given by David and Journe providesa general criterion for theL2.Rn/-boundedness of generalized Calderon-Zygmundsingular integral operators; see [5, 35]. The T1 theorem, however, cannot be directly

The first author was supported by NNSF (No. 10171111) of China and the second (corresponding)author was supported by NNSF (No. 10271015) and RFDP (No. 20020027004) of China. The authorsalso want to thank the referee for their several helpful comments which made this article more readable.c© 2006 Australian Mathematical Society 1446-8107/06$A2:00+ 0:00

229

http://www.austms.org.au/Publ/JAustMS/V80P2/q143.html

230 Dongguo Deng and Dachun Yang [2]

applied to the Cauchy integral on Lipschitz curves. Meyer in [30] (see also [31])observed that if the function 1 in theT1 theorem is allowed to be replaced by abounded complex-valued functionb satisfying 0< Ž ≤ Reb.x/ almost everywhere,then this result would imply theL2.Rn/ boundedness of the Cauchy integral on allLipschitz curves. Replacing the function 1 by an accretive function, McIntosh andMeyer in [30] proved theT b theorem, whereb is an accretive function. David, Journe,and Semmes in [6] introduced a more general class ofL∞.Rn/ functionsb, namely,the so-called para-accretive functions. They proved that the function 1 in theT1theorem can be replaced by para-accretive functions, which is why it is now calledthe T b theorem. Moreover, they showed that the para-accretivity is also necessaryin the sense that if theT b theorem holds for a bounded functionb, thenb is para-accretive. Moreover, Meyer in [31] observed that ifb.x/ is a bounded function and1 ≤ Reb.x/, one can then define the modified Hardy spaceH1

b .Rn/ simply via the

classical Hardy spaceH1.Rn/, that is, the spaceH1b .R

n/ is defined by the collectionof all functions f such thatbf is in the classical Hardy spaceH1.Rn/. This spacehas the advantage of the cancellation adapted to the complex measureb.x/ dx andis closely related to theT b theorem, whereb is an accretive function. In fact, Han,Lee and Lin recently proved in [17] that if T∗.b/ = 0, then the Calderon-ZygmundoperatorT is bounded from the classicalH p.Rn/ to a new Hardy spaceH p

b .Rn/ for

n=.n + ž/ 1, the Besov spaces,bBs

pq.X/ andb−1Bspq.X/, the Triebel-Lizorkin spaces,

bFspq.X/ andb−1Fs

pq.X/, were considered by Han in [14], and the relatedT b theoremwas also established in that paper.

The main purpose of this paper is to study the Besov spaces,bBspq.X/ and

b−1Bspq.X/, the Triebel-Lizorkin spaces,bFs

pq.X/ and b−1Fspq.X/ when p ≤ 1 or

q ≤ 1, and the relatedT b Theorem. We will do this in the setting of metric spaces, ormore generally, spaces of homogeneous type.

Analysis on metric spaces has recently obtained an increasing interest; see [34,25, 13, 27]. In particular, the theory of function spaces on metric spaces, or moregenerally, the spaces of homogeneous type in the sense of Coifman and Weiss in[3, 4] has been well developed; see [29, 28, 22, 16, 19, 20, 21, 23, 24, 42, 48]. It iswell known that the spaces of homogeneous type in Definition1.1below include theEuclidean space, theC∞ Riemannian manifolds, the boundaries of Lipschitz domains,and, in particular, the Lipschitz manifolds introduced recently by Triebel in [41] andthe isotropic and anisotropicd-sets inRn. It has been proved by Triebel in [39] thatthed-sets inRn include various kinds of self-affine fractals, for example, the Cantorset, the generalized Sierpinski carpet and so forth. We point out that the spaces ofhomogeneous type in Definition1.1also include the post critically finite self-similarfractals studied by Kigami in [26] and by Strichartz in [36], and the metric spaces with

[3] Besov and Triebel-Lizorkin spaces 231

heat kernel studied by Grigor’yan, Hu and Lau in [12]. More examples of spaces ofhomogeneous type can be found in [3, 4, 34].

We now state some necessary definitions and notation of spaces of homogeneoustype. A quasi-metric² on a setX is a function² : X × X → [0;∞/ satisfyingthat

(i) ².x; y/ = 0 if and only if x = y;(ii) ².x; y/ = ².y; x/ for all x; y ∈ X;(iii) there exists a constantA ∈ [1;∞/ such that for allx; y andz ∈ X,

².x; y/ ≤ A[².x; z/+ ².z; y/]:

Any quasi-metric defines a topology, for which the ballsB.x; r / = {y ∈ X :².y; x/ < r } for all x ∈ X andr > 0 form a basis.

In what follows, we set diamX = sup{².x; y/ : x; y ∈ X}. We also assume thefollowing conventions. We denote byf ∼ g that there is a constantC > 0 independentof the main parameters such thatC−1g < f < Cg. Throughout the paper,C willdenote a positive constant which is independent of the main parameters, however itmay vary from line to line. Constants with subscripts, such asC1, do not change indifferent occurrences. For anyq ∈ [1;∞], we denote byq′ its conjugate index, thatis, 1=q +1=q′ = 1. Let A be a set and we will denote by�A the characteristic functionof A.

DEFINITION 1.1 ([23]). Let d > 0 and� ∈ .0; 1]. A space of homogeneous type,.X; ²; ¼/d;� , is a setX, together with a quasi-metric², a nonnegative Borel regularmeasure¼ on X, and, in addition, there exists a constantC0 > 0 such that for all0< r < diamX andx; x′; y ∈ X,

¼.B.x; r // ∼ r d(1.1)

and

|².x; y/− ².x′; y/| ≤ C0².x; x′/� [².x; y/+ ².x′; y/]1−� :(1.2)

The space of homogeneous type defined above is an variant of the space of homo-geneous type introduced by Coifman and Weiss in [3]. In [29], Macias and Segoviahave proved that one can replace the quasi-metric² of the space of homogeneous typein the sense of Coifman and Weiss by another quasi-metric² which yields the sametopology onX as² such that.X; ²; ¼/ is the space defined by Definition1.1 withd = 1.

Throughout this paper, we will always assume that¼.X/ = ∞.Let us now recall the definitions of para-accretive functions and the space of test

functions.


DEFINITION 1.2. A bounded complex-valued functionb on X, a space of homoge-neous type, is said to bepara-accretiveif there exist constantsC1 > 0 and� ∈ .0; 1]such that for all ballsB ⊂ X, there is a ballB′ ⊂ B with �¼.B/ ≤ ¼.B′/ satisfying

1

¼.B/

∣∣∣∣∫

B′b.x/ d¼.x/

∣∣∣∣ ≥ C1 > 0:

DEFINITION 1.3 ([14]). Letb be a para-accretive function. Fix >0 and� ≥þ >0.A function f defined onX is said to be atest functionof type.x0; r; þ; /with x0 ∈ Xandr > 0, if f satisfies the following conditions:

(i) | f .x/| ≤ Cr

.r + ².x; x0//d+ ;

(ii) | f .x/ − f .y/| ≤ C

(².x; y/

r + ².x; x0/

)þ r

.r + ².x; x0//d+ for ².x; y/ ≤[r + ².x; x0/]=2A;

(iii)∫

X

f .x/b.x/ d¼.x/ = 0:

If f is a test function of type.x0; r; þ; / related to a para-accretive functionb, wewrite f ∈ Gb.x0; r; þ; / and the norm off in Gb.x0; r; þ; / is defined by

‖ f ‖Gb.x0;r;þ; / = inf{C : (i) and (ii) hold}:Now fix x0 ∈ X and letGb.þ; / = Gb.x0; 1; þ; /. It is easy to see that

Gb.x1; r; þ; / = Gb.þ; /

with an equivalent norm for allx1 ∈ X andr > 0. Furthermore, it is easy to check thatGb.þ; / is a Banach space with respect to the norm inGb.þ; /. Also, let the dualspace.Gb.þ; //

′ be all linear functionalsL from Gb.þ; / to C with the propertythat there exists aC ≥ 0 such that for allf ∈ Gb.þ; /,

|L . f /| ≤ C‖ f ‖Gb.þ; /:

We denote by〈h; f 〉 the natural pairing of elementsh ∈ .Gb.þ; //′ and f ∈ Gb.þ; /.

Clearly, for allh ∈ .Gb.þ; //′, 〈h; f 〉 is well defined for all f ∈ Gb.x0; r; þ; / with

x0 ∈ X andr > 0.It is well known that even whenX = Rn, Gb.þ1; / is not dense inGb.þ2; / if

þ1 > þ2, which will bring us some inconvenience. To overcome this defect, in whatfollows, for a givenž ∈ .0; � ], we letGb.þ; / be the completion of the spaceGb.ž; ž/

in Gb.þ; / when 0< þ, < ž.Let b be a para-accretive function. As usual, we write

bGb.þ; / = { f : f = bg for some g ∈ Gb.þ; /}:


If f ∈ bGb.þ; / and f = bg for someg ∈ Gb.þ; /, then the norm off is definedby ‖ f ‖bGb.þ; / = ‖g‖Gb.þ; /. By this definition, it is easy to see that

f ∈ (bGb.þ; /)′

if and only if bf ∈ (Gb.þ; /)′;(1.3)

where we definebf ∈ (Gb.þ; /)′

by 〈bf; g〉 = 〈 f; bg〉 for all g ∈ Gb.þ; /.

DEFINITION 1.4 ([16]). Let b be a para-accretive function. A sequence{Sk}bk∈Z of

linear operators is said to be anapproximation to the identityof order ž ∈ .0; � ]associated tob if there existsC2 > 0 such that for allk ∈ Z and allx, x′, y, y′ ∈ X,the kernel ofSk, denoted bySk.x; y/, is a function fromX × X intoC satisfying

(i) |Sk.x; y/| ≤ C22−kž

.2−k + ².x; y//d+ž ;

(ii) |Sk.x; y/− Sk.x′; y/| ≤ C2

(².x; x′/

2−k + ².x; y/

)ž 2−kž

.2−k + ².x; y//d+ž for

².x; x′/ ≤ .2−k + ².x; y//=2A;

(iii) |Sk.x; y/− Sk.x; y′/| ≤ C2

(².y; y′/

2−k + ².x; y/

)ž 2−kž

.2−k + ².x; y//d+ž for

².y; y′/ ≤ .2−k + ².x; y//=2A;(iv)

∣∣[Sk.x; y/− Sk.x; y′/] − [Sk.x′; y/− Sk.x

′; y′/]∣∣≤ C2

(².x; x′/

2−k + ².x; y/

)ž (².y; y′/

2−k + ².x; y/

)ž 2−kž

.2−k + ².x; y//d+ž

for ².x; x′/ ≤ .2−k + ².x; y//=2A and².y; y′/ ≤ .2−k + ².x; y//=2A;

(v)∫

X

Sk.x; y/b.y/ d¼.y/ = 1;

(vi)∫

X

Sk.x; y/b.x/ d¼.x/ = 1:

REMARK 1.5. By Coifman’s construction in [6], if b is a given para-accretivefunction, one can construct an approximation to the identity of order� such thatSk.x; y/ has compact support when one variable is fixed, that is, there is a constantC3 > 0 such that for allk ∈ Z, Sk.x; y/ = 0 if ².x; y/ ≥ C32−k.

REMARK 1.6. We also remark that in the sequel, if the approximation to the identityas in Definition1.2exists, then all the results still hold whenb andb−1 are bounded.It seems that we do not need to assume thatb is a para-accretive function. However,in [6], it was proved that the existence of the approximation to the identity as inDefinition1.2 is equivalent to the para-accretivity ofb.

In the next section, we introduce the norms‖ · ‖bBspq.X/

and‖ · ‖bFspq.X/

on somedistribution spaces via approximations to the identity. Then we establish a Plancherel-


Polya-type inequality related to these norms by using the discrete Calderon repro-ducing formula in [17]. From this inequality, we deduce that the definitions of thesenorms are independent of the choice of approximations to the identity. By the discreteCalderon reproducing formula again, we further verify that the definition of thesenorms are also independent of the choice of the distribution spaces under some re-strictions. After these preparations, we introduce the Besov spacebBs

pq.X/ and theTriebel-Lizorkin spacebFs

pq.X/. Using the discrete Calderon reproducing formula,we then establish the frame and the Littlewood-Paley function characterizations ofthese spaces. By the results in [6] and [17], it is easy to see thatbF0

p2.X/ = L p.X/with an equivalent norm if 1< p < ∞, andbF0

p2.X/ = H p.X/ with an equivalentnorm if d=.d + ž/ < p ≤ 1. It is still unknown if this is true for any others; p andq.

In Section3 of this paper, we introduce the new Besov spaceb−1Bspq.X/ and the

new Triebel-Lizorkin spaceb−1Fspq.X/. The frame and the Littlewood-Paley function

characterization of these spaces are also given. Moreover, the relations among thespacesbBs

pq.X/, bFspq.X/, b−1Bs

pq.X/, b−1Fspq.X/, and the known Hardy spaces are

presented.Section4 is devoted to some applications of the theory of these spaces. Using

the frame characterization of these spaces, we first establish some real interpolationtheorems. Some embedding theorems on these spaces are also presented. Usingthe interpolation theorem, we further establish theT b theorems on these spaces andconsider the boundedness of new Riesz potentials in these spaces. As a corollary,we obtain the boundedness of the new Riesz potential operatorIÞ for 0 ≤ Þ < ž

with ž ∈ .0; � ] from the Hardy spacesH p.X/ to the Hardy spacesH qb .X/, where

1=q = 1=p−Þ=d andd=.d+ž−Þ/ < p < ∞. Moreover, using theT bTheorem, wefurther establish the lifting property via the Riesz potential operators of these spaces.

Finally, we mention that the theory of the spacesbFs∞q.X/ andb−1Fs

∞q.X/ with|s| < ž ∈ .0; � ] and max{d=.d + ž/; d=.d + s + ž/} < q ≤ ∞ has been establishedin [48].

2. SpacesbBspq(X) and bF s

pq(X)

Let b be a para-accretive function as in Definition1.2. We now introduce the norms‖ · ‖bBs

pq.X/and‖ · ‖bFs

pq.X/on some distribution spaces via the approximations to the

identity. In what follows, we denote max.0; x/ for anyx ∈ R simply byx+.

DEFINITION 2.1. Let b be a para-accretive function as in Definition1.2. Let ž ∈.0; � ], {Sk}b

k∈Z be an approximation to the identity of orderž as in Definition1.4, andDk = Sk − Sk−1 for k ∈ Z. Let |s| < ž.

(i) If max{d=.d + ž/; d=.d + s + ž/} < p ≤ ∞ and 0< q ≤ ∞, for all


f ∈ (Gb.þ; /)′

with

max{0;−s + d.1=p − 1/+} < þ < ž and max{0; s − d=p} < < ž;(2.1)

we define the norm‖ f ‖bBspq.X/

by

‖ f ‖bBspq.X/

={ ∞∑

k=−∞2ksq‖Dk. f /‖q

L p.X/

}1=q

with the usual modifications made whenp = ∞ or q = ∞.(ii) If max{d=.d+ž/; d=.d+s+ž/} < p < ∞and max{d=.d+ž/; d=.d+s+ž/} <

q ≤ ∞, for all f ∈ (Gb.þ; /)′

with þ; as in (2.1), we define the norm‖ f ‖bFspq.X/

by

‖ f ‖bFspq.X/

=∥∥∥∥∥∥{ ∞∑

k=−∞2ksq|Dk. f /|q

}1=q∥∥∥∥∥∥

L p.X/

with the usual modification made whenq = ∞.

The following theorem indicates that the definitions of the norms‖ · ‖bBspq.X/

and‖ · ‖bFs

pq.X/are independent of the choice of approximations to the identity.

THEOREM 2.2. Let b be a para-accretive function as in Definition1.2, ž ∈ .0; � ]and|s| < ž. Let{Sk}b

k∈Z and{Gk}bk∈Z be two approximations to the identity of orderž

as in Definition1.4, Dk = Sk − Sk−1 and Ek = Gk − Gk−1 for k ∈ Z.

(i) If max{d=.d + ž/; d=.d + s + ž/} < p ≤ ∞ and 0 < q ≤ ∞, for allf ∈ (Gb.þ; /

)′with 0< þ; < ž, we have

{ ∞∑k=−∞

2ksq‖Dk. f /‖qL p.X/

}1=q

∼{ ∞∑

k=−∞2ksq‖Ek. f /‖q

L p.X/

}1=q

:

(ii) If max{d=.d + ž/; d=.d + s+ ž/} < p < ∞ andmax{d=.d + ž/; d=.d + s+ž/} < q ≤ ∞, for all f ∈ (Gb.þ; /

)′with 0< þ; < ž, we have∥∥∥∥∥∥

{ ∞∑k=−∞

2ksq|Dk. f /|q

}1=q∥∥∥∥∥∥

L p.X/

∼∥∥∥∥∥∥{ ∞∑

k=−∞2ksq|Ek. f /|q

}1=q∥∥∥∥∥∥

L p.X/

:

To prove Theorem2.2, we first recall the following construction given by Christin [2], which provides an analogue of the grid of Euclidean dyadic cubes on spaces ofhomogeneous type.


LEMMA 2.3. Let X be a space of homogeneous type. Then there exists a collection{Qk

Þ ⊂ X : k ∈ Z; Þ ∈ Ik} of open subsets, whereIk is some index set,Ž ∈ .0; 1/ andC4; C5 > 0, such that

(i) ¼.X \ ∪ÞQkÞ/ = 0 for each fixedk and Qk

Þ ∩ Qkþ = ∅ if Þ 6= þ;

(ii) for anyÞ; þ; k; l with l ≥ k; either Qlþ ⊂ Qk

Þ or Qlþ ∩ Qk

Þ = ∅;(iii) for each.k; Þ/ and eachl < k there is a uniqueþ such thatQk

Þ ⊂ Qlþ ;

(iv) diam.QkÞ/ ≤ C4Ž

k;(v) eachQk

Þ contains some ballB.zkÞ;C5Ž

k/, wherezkÞ ∈ X.

In fact, we can think ofQkÞ as being a dyadic cube with diameter roughŽk and

centered atzkÞ: In what follows, we always supposeŽ = 1=2: See [22] for how to

remove this restriction. In addition, fork ∈ Z and− ∈ Ik, we will denote byQk;¹− ;

¹ = 1; 2; : : : ; N.k; − /; the set of all cubesQk+ j− ′ ⊂ Qk

− ; where j is a fixed largepositive integer. Denote byyk;¹

− a point inQk;¹− :

REMARK 2.4. Since we always assume that¼.X/ = ∞ in this paper, for allk ∈ Z,Ik in Lemma2.3 is an infinite index set.

Theorem2.2 is a simple corollary of Lemma2.3 and the following Plancherel-Polya-type inequality.




(i) If max{d=.d + ž/; d=.d + s + ž/} 0 such that for all f ∈ (Gb.þ; /

)′with 0< þ; < ž, we have

∞∑k=−∞

2ksq

[∑−∈Ik

N.k;− /∑¹=1

¼.Qk;¹− / sup

z∈Qk;¹−

|Dk. f /.z/|p

]q=p

1=q

≤ C

∞∑k=−∞

2ksq

[∑−∈Ik

N.k;− /∑¹=1

¼.Qk;¹− / inf

z∈Qk;¹−

|Ek. f /.z/|p

]q=p

1=q

:

(ii) If max{d=.d + ž/; d=.d + s + ž/} 0 such that for all f ∈ (

Gb.þ; /)′

with0< þ; < ž, we have∥∥∥∥∥∥

{ ∞∑k=−∞

∑−∈Ik

N.k;− /∑¹=1

2ksq supz∈Qk;¹

−

|Dk. f /.z/|q�Qk;¹−

}1=q∥∥∥∥∥∥

L p.X/


≤ C

∥∥∥∥∥∥{ ∞∑

k=−∞

∑−∈Ik

N.k;− /∑¹=1

2ksq infz∈Qk;¹

−

|Ek. f /.z/|q�Qk;¹−

}1=q∥∥∥∥∥∥

L p.X/

:

To show Theorem2.5, the following discrete Calderon reproducing formula in [17]will play a crucial role.

LEMMA 2.6. Letb be a para-accretive function as in Definition1.2. Letž ∈ .0; � ],{Gk}b

k∈Z be an approximation to the identity of orderž as in Definition1.4, andEk = Gk − Gk−1 for k ∈ Z. Let0< þ; < ž. Then there exists a family of functions{Ek.x; y/}k∈Z such that for all fixedyk;¹

− ∈ Qk;¹− and all f ∈ (Gb.þ; /

)′,

f .x/ =∑k∈Z

∑−∈Ik

N.k;− /∑¹=1

Ek. f /.yk;¹− /

∫Qk;¹−

b.x/Ek.y; x/b.y/ d¼.y/;(2.2)

where Ek.x; y/ for k ∈ Z satisfies(i) and (iii) of Definition1.4 with ž replaced byž′ ∈ .0; ž/, and∫

X

Ek.x; y/b.x/ d¼.x/ =∫

X

Ek.x; y/b.y/ d¼.y/ = 0:

Moreover, the series in(2.2) converges in the sense that for allg ∈ Gb.þ′; ′/ with

þ < þ ′ < ž and < ′ < ž,

limM; N→∞

⟨∑|k|≤M

∑−∈Ik

².x0;zk− /≤N

N.k;− /∑¹=1

Ek. f /.yk;¹− /

∫Qk;¹−

b.·/Ek.y; ·/b.y/ d¼.y/; g

⟩= 〈 f; g〉:

Another useful tool in dealing with the Triebel-Lizorkin space is the followinglemma which can be found in [8, pages 147–148] forRn and [22, page 93] for spacesof homogeneous type.

LEMMA 2.7. Let 0 < r ≤ 1, k; � ∈ Z+ with � ≤ k and for any dyadic cubeQk;¹−

and all x ∈ X, | fQk;¹−.x/| ≤ .1+2�².x; yk;¹

− //−d− , whereyk;¹

− is any point inQk;¹− and

> d.1=r − 1/. Then, for allx ∈ X,

∑−∈Ik

N.k;− /∑¹=1

|½Qk;¹−

|| fQk;¹−.x/| ≤ C 2.k−�/d=r

[M

(∑−∈Ik

N.k;− /∑¹=1

|½Qk;¹−

|r�Qk;¹−

).x/

]1=r

;

whereC is independent ofx, k and �, and M is the Hardy-Littlewood maximaloperator onX; see[3].


PROOF OFTHEOREM2.5. We first verify (i). With all the notation as in Theorem2.5,by (2.2), we have that for allk ∈ Z andz ∈ X,

Dk. f /.z/ =∞∑

k′=−∞

∑− ′∈Ik′

N.k′;− ′/∑¹′=1

Ek′. f /.yk′;¹′− ′ /

∫Qk′ ;¹′− ′

Dk.bEk′.y; ·//.z/b.y/ d¼.y/:(2.3)

We recall the estimate in [14, page 66], that there is a constantC > 0 such that for allk; k′ ∈ Z and ally; z ∈ X,

∣∣∣Dk.bEk′.y; ·//.z/∣∣∣ ≤ Cž′2−|k−k′|ž′ 2−.k∧k′/ž′

.2−.k∧k′/ + ².y; z//d+ž′ ;(2.4)

wherež′ can be any positive number in.0; ž/ andk ∧ k′ = min.k; k′/.If y ∈ Qk′;¹′

− ′ andz ∈ Qk;¹− , then

2−.k∧k′/ + ².y; z/ ∼ 2−.k∧k′/ + ².yk′;¹′− ′ ; yk;¹

− /:(2.5)

We also recall the well-known inequality that for allaj ∈ C andq ≤ 1,(∑j

|aj |)q

≤∑

j

|aj |q:(2.6)

Now supposep ≤ 1. In this case, from (2.3)–(2.6), it follows that

∞∑k=−∞

2ksq

[∑−∈Ik

N.k;− /∑¹=1

¼.Qk;¹− / sup

z∈Qk;¹−

|Dk. f /.z/|p

]q=p

1=q

≤ C

∞∑k=−∞

2ksq

(∑−∈Ik

N.k;− /∑¹=1

¼.Qk;¹− /

[ ∞∑k′=−∞

∑− ′∈Ik′

N.k′;− ′/∑¹′=1

2−|k−k′ |ž′ p¼.Qk′;¹′− ′ /

p

× ∣∣Ek′. f /.yk′;¹′− ′ /

∣∣p 2−.k∧k′/ž′ p

.2−.k∧k′/ + ².yk′;¹′− ′ ; yk;¹

− //.d+ž′/p

])q=p

1=q

≤ C

∞∑k=−∞

[ ∞∑k′=−∞

2.k−k′/sp−|k−k′|ž′ p+k′d.1−p/∑− ′∈Ik′

N.k′;− ′/∑¹′=1

2k′sp¼.Qk′;¹′− ′ /

× ∣∣Ek′. f /.yk′;¹′− ′ /

∣∣p∫

X

2−.k∧k′/ž′ p

.2−.k∧k′/ + ².yk′;¹′− ′ ; x//.d+ž′/p

d¼.x/

]q=p

1=q

≤ C

∞∑k=−∞

[ ∞∑k′=−∞

2.k−k′/sp−|k−k′|ž′ p−.k∧k′/d.1−p/+k′d.1−p/


×(∑− ′∈Ik′

N.k′;− ′/∑¹′=1

2k′sp¼.Qk′;¹′− ′ /

∣∣Ek′. f /.yk′;¹′− ′ /

∣∣p

)]q=p

1=q

≤ C

∞∑k′=−∞

[∑− ′∈Ik′

N.k′;− ′/∑¹′=1

2k′sp¼.Qk′;¹′− ′ /

∣∣Ek′. f /.yk′;¹′− ′ /

∣∣p

]q=p

1=q

;

where we choosež′ ∈ .0; ž/ such thatž′ > max{s;−s + d.1=p − 1/}, and in the lastinequality, ifq=p ≥ 1, we use the Holder inequality and the following estimate( ∞∑

k′=−∞+

∞∑k=−∞

)2.k−k′/sp−|k−k′|ž′ p−.k∧k′/d.1−p/+k′d.1−p/ ≤ C:

If q=p < 1, we use (2.6) and∞∑

k=−∞2[.k−k′/sp−|k−k′|ž′ p−.k∧k′/d.1−p/+k′d.1−p/]q=p ≤ C:

Sinceyk′;¹′− ′ is arbitrary, we can further obtain a desired estimate in this case.

If p > 1, (2.3)–(2.5) and the Holder inequality yield that

∞∑k=−∞

2ksq

[∑−∈Ik

N.k;− /∑¹=1

¼.Qk;¹− / sup

z∈Qk;¹−

|Dk. f /.z/|p

]q=p

1=q

≤ C

∞∑k=−∞

[∑−∈Ik

N.k;− /∑¹=1

¼.Qk;¹− /

( ∞∑k′=−∞

2.k−k′/sp−|k−k′|ž′ p

×[∑− ′∈Ik′

N.k′;− ′/∑¹′=1

2k′sp¼.Qk′;¹′− ′ /

∣∣Ek′. f /.yk′;¹′− ′ /

∣∣p 2−.k∧k′/ž′

.2−.k∧k′/ + ².yk′;¹′− ′ ; yk;¹

− //d+ž′

]

×[∫

X

2−.k∧k′/ž′

.2−.k∧k′/ + ².y; yk;¹− //

d+ž′ d¼.y/

]p=p′)]q=p

1=q

≤ C

∞∑k=−∞

[ ∞∑k′=−∞

2.k−k′/sp−|k−k′|ž′ p

×(∑− ′∈Ik′

N.k′;− ′/∑¹′=1

2k′sp¼.Qk′;¹′− ′ /

∣∣Ek′. f /.yk′;¹′− ′ /

∣∣p

)

×∫

X

2−.k∧k′/ž′

.2−.k∧k′/ + ².yk′;¹′− ′ ; x//d+ž′ d¼.x/

]q=p

1=q


≤ C

∞∑k′=−∞

[∑− ′∈Ik′

N.k′;− ′/∑¹′=1

2k′sp¼.Qk′;¹′− ′ /

∣∣Ek′. f /.yk′;¹′− ′ /

∣∣p

]q=p

1=q

;

where we choosež′ > |s| and we use the fact that for allx ∈ X,∫X

2−.k∧k′/ž′

.2−.k∧k′/ + ².y; x//d+ž′ d¼.y/ ≤ C;( ∞∑k′=−∞

+∞∑

k=−∞

)2.k−k′/sp−|k−k′|ž′ p ≤ C;

and∞∑

k=−∞2[.k−k′/sp−|k−k′|ž′ p]q=p ≤ C:

Sinceyk′;¹′− ′ is arbitrary, a desired estimate follows, and this finishes the proof of (i).

To verify (ii), by Lemma2.6, Lemma2.7, (2.4)–(2.6) or the Holder inequality, wehave{ ∞∑

k=−∞

∑−∈Ik

N.k;− /∑¹=1

2ksq supz∈Qk;¹

−

|Dk. f /.z/|q�Qk;¹−.x/

}1=q

≤ C

{ ∞∑k=−∞

2ksq

[ ∞∑k′=−∞

∑− ′∈Ik′

N.k′;− ′/∑¹′=1

2−|k−k′|ž′¼.Qk′;¹′

− ′ /∣∣Ek′. f /.yk′;¹′

− ′ /∣∣

× 2−.k∧k′/ž′

.2−.k∧k′/ + ².x; yk′;¹′− ′ //d+ž′

]q}1=q

≤ C

∞∑k=−∞

∞∑

k′=−∞2.k−k′/s−|k−k′|ž′+[.k∧k′/−k′]d.1−1=r /

×[

M

(∑− ′∈Ik′

N.k′;− ′/∑¹′=1

2k′sr∣∣Ek′. f /.yk′;¹′

− ′ /∣∣r�Qk′ ;¹′

− ′

).x/

]1=r

q

1=q

≤ C

∞∑k′=−∞

[M

(∑− ′∈Ik′

N.k′;− ′/∑¹′=1

2k′sr∣∣Ek′. f /.yk′;¹′

− ′ /∣∣r�Qk′ ;¹′

− ′

).x/

]q=r

1=q

;

where we choosež′ ∈ .0; ž/ andr ∈ .0; 1] such thatž′ > max{d.1=r − 1/; s;−s +d.1=r − 1/} andr < min{p;q}, and we use the fact that( ∞∑

k′=−∞+

∞∑k=−∞

)2.k−k′/s−|k−k′|ž′+[.k∧k′/−k′]d.1−1=r / ≤ C;


and∑∞

k=−∞ 2.k−k′/s−|k−k′|ž′+[.k∧k′/−k′]d.1−1=r /q ≤ C. Thus, the vector-valued inequalityof Fefferman and Stein in [7] and the arbitrariness ofyk′;¹′

− ′ ∈ Qk′;¹′− ′ further imply the

conclusion (ii) of the theorem, which completes the proof of Theorem2.5.

REMARK 2.8. From the proof of Theorem2.5, it is easy to see that the key roleplayed by{Dk}k∈Z is the estimate of (2.4). However, to establish this estimate, weneed only to use the regularity (iii) as in Definition1.4of Dk for k ∈ Z; see [14, 16].This means that if we replace the operatorsDk by some other operatorsDk for k ∈ Zwhose kernels have the same properties as the kernels ofDk except for the regularity(ii) of Definition 1.4, then Theorem2.5still holds. This observation is useful in someapplications.

Let us now verify that under some restrictions onþ and , the definition of thenorms‖ · ‖bBs

pq.X/and‖ · ‖bFs

pq.X/is independent of the choice of the distribution space,(

G .þ; /)′

.

THEOREM 2.9. Let b be a para-accretive function as in Definition1.2, ž ∈ .0; � ]and|s| < ž.

(i) Let max{d=.d + ž/; d=.d + s + ž/} < p ≤ ∞ and 0 < q ≤ ∞. If f ∈(Gb.þ1; 1/

)′with max{0;−s+ d.1=p − 1/+} < þ1 < ž, max{0; s − d=p} < 1 < ž,

and‖ f ‖bBspq.X/

< ∞, then f ∈ (Gb.þ2; 2/)′

withmax{0;−s+d.1=p−1/+} < þ2 < ž

andmax{0; s − d=p} < 2 < ž.(ii) Let

max

{d

d+ž ;d

d+s+ž}< p < ∞ and max

{d

d+ž ;d

d+s+ž}< q ≤ ∞:

If f ∈ (Gb.þ1; 1/)′

with max{0;−s + d.1=p − 1/+} < þ1 < ž, max{0; s − d=p} < 1 < ž, and‖ f ‖bFs

pq.X/< ∞, then f ∈ (Gb.þ2; 2/

)′withmax{0;−s+d.1=p−1/+} <

þ2 < ž andmax{0; s − d=p} < 2 < ž.

PROOF. Let ∈ Gb.ž; ž/. With all the notation as in Lemma2.6, we first claim fork = 0; 1; 2; : : : , − ∈ Ik and¹ = 1; 2; : : : ; N.k; − / that

∣∣〈b.·/Ek.y; ·/; 〉∣∣ ≤ C 2−kþ2‖ ‖Gb.þ2; 2/

1

.1 + ².y; x0//d+ 2;(2.7)

and fork = −1;−2; : : : , − ∈ Ik and¹ = 1; 2; : : : ; N.k; − / that

∣∣〈b.·/Ek.y; ·/; 〉∣∣ ≤ C 2k ′2‖ ‖Gb.þ2; 2/

2−k 2

.2−k + ².y; x0//d+ 2;(2.8)

where ′2 can be any positive number in.0; 2/.


In fact, for (2.7), by the vanishing moment ofb.·/Ek.y; ·/, we have∣∣⟨b.·/Ek.y; ·/; ⟩∣∣

=∣∣∣∣∫

X

Ek.y; x/b.x/[ .x/− .y/] d¼.x/

∣∣∣∣≤ C‖ ‖Gb.þ2; 2/

{∫².y;x/≤.1+².y;x0//=2A

|Ek.y; x/| ².x; y/þ2

.1 + ².y; x0//d+ 2+þ d¼.x/

+∫².y;x/>.1+².y;x0//=2A

|Ek.y; x/|[

1

.1+².x; x0//d+ 2+ 1

.1+².y; x0//d+ 2

]d¼.x/

}

≤ C 2−kþ2‖ ‖Gb.þ2; 2/

1

.1 + ².y; x0//d+ 2;

where we choosež ′ > 0 in Lemma2.6such thatž ′ > max{ 2; þ2}. This verifies (2.7).Similarly, the vanishing moment ofb yields that∣∣⟨b.·/Ek.y; ·/;

⟩∣∣=∣∣∣∣∫

X

[Ek.y; x/− Ek.y; x0/

]b.x/ .x/ d¼.x/

∣∣∣∣≤ C‖ ‖Gb.þ2; 2/

{∫².x;x0/≤.2−k+².y;x0//=2A

2−kž′².x; x0/

′2

.2−k +².y; x0//d+ž′+ ′2

d¼.x/

.1+².x; x0//d+ 2

+∫².x;x0/>.2−k+².y;x0//=2A

[2−kž′

.2−k + ².y; x//d+ž′ + 2−kž′

.2−k + ².y; x0//d+ž′

]

× 1

.1 + ².x; x0//d+ 2d¼.x/

}

≤ C 2k ′2‖ ‖Gb.þ2; 2/

2−k 2

.2−k + ².y; x0//d+ 2;

which is just (2.8).Observe that ifk ≥ 0, − ∈ Ik, ¹ = 1; : : : ; N.k; − /, andy ∈ Qk;¹

− , then

1 + ².y; x0/ ∼ 1 + ².yk;¹− ; x0/;(2.9)

and if k = −1;−2; : : : , − ∈ Ik, ¹ = 1; : : : ; N.k; − /, andy ∈ Qk;¹− , then

2−k + ².y; x0/ ∼ 2−k + ².yk;¹− ; x0/:(2.10)

The estimates (2.7) and (2.8) and the observations (2.9) and (2.10) tell us that

|〈 f; 〉| =∣∣∣∣∣

∞∑k=−∞

∑−∈Ik

N.k;− /∑¹=1

Ek. f /.yk;¹− /

∫Qk;¹−

⟨b.·/Ek.y; ·/;

⟩b.y/ d¼.y/

∣∣∣∣∣


≤ C‖ ‖Gb.þ2; 2/

{ ∞∑k=0

∑−∈Ik

N.k;− /∑¹=1

2−kþ2¼.Qk;¹− /|Ek. f /.yk;¹

− /|

× 1

.1 + ².yk;¹− ; x0//d+ 2

+−1∑

k=−∞

∑−∈Ik

N.k;− /∑¹=1

2k ′2¼.Qk;¹

− /|Ek. f /.yk;¹− /|

2−k 2

.2−k + ².yk;¹− ; x0//d+ 2

}:

If p ≤ 1, Lemma2.3, Theorem2.5, (2.6) and the Holder inequality yield that

|〈 f; 〉| ≤ C‖ ‖Gb.þ2; 2/

∞∑k=0

2−k.þ2+s/−kd.1−1=p/(2.11)

×(∑−∈Ik

N.k;− /∑¹=1

2ksp¼.Qk;¹− /|Ek. f /.yk;¹

− /|p

)1=p

+−1∑

k=−∞2k. ′

2−s+d=p/

(∑−∈Ik

N.k;− /∑¹=1

2ksp¼.Qk;¹− /|Ek. f /.yk;¹

− /|p

)1=p

≤ C‖ ‖Gb.þ2; 2/

{ ∞∑k=−∞

2ksq‖Ek. f /‖qL p.X/

}1=q

≤ C‖ ‖Gb.þ2; 2/‖ f ‖bBspq.X/

;

where we choose ′2 ∈ .0; 2/ such that ′

2 > s − d=p and use the fact that

þ2 > −s + d.1=p − 1/:

If p > 1, similar estimates, except that (2.4) is replaced by the Holder inequality,lead us to

|〈 f; 〉| ≤ C‖ ‖Gb.þ2; 2/

∞∑k=0

2−kþ2

[∑−∈Ik

N.k;− /∑¹=1

¼.Qk;¹− /|Ek. f /.yk;¹

− /|p

]1=p

(2.12)

×[∫

X

1

.1 + ².y; x0//d+ 2d¼.y/

]1=p′

+−1∑

k=−∞2k. ′

2+d=p/

[∑−∈Ik

N.k;− /∑¹=1

¼.Qk;¹− /|Ek. f /.yk;¹

− /|p

]1=p

×[∫

X

2−k 2

.2−k + ².y; x0//d+ 2d¼.y/

]1=p′ ≤ C‖ ‖Gb.þ2; 2/‖ f ‖bBs

pq.X/;


where we use the fact thatþ2 > −s in this case and choose ′2 > max{0; s − d=p}.

Supposeh ∈ G .þ2; 2/. We choosehn ∈ G .ž; ž/ for anyn ∈ N such that

‖hn − h‖G .þ2; 2/

n→∞−−→ 0:

The estimates of (2.11) and (2.12) show that for alln; m ∈ N,

|〈 f; hn − hm〉| ≤ C‖ f ‖bBspq.X/

‖hn − hm‖G .þ2; 2/;

which shows limn→∞〈 f; hn〉 exists and the limit is independent of the choice ofhn.Therefore, we define

〈 f; h〉 = limn→∞

〈 f; hn〉:

By (2.11) and (2.12), for all h ∈ G .þ2; 2/,

|〈 f; h〉| ≤ C‖ f ‖bBspq.X/

‖h‖G .þ2; 2/:

Thus, f ∈ (G .þ2; 2/)′

. This finishes the proof of (i).The conclusion (ii) can be deduced from (i) and the fact that

‖ f ‖bBsp;max.p;q/.X/

≤ C‖ f ‖bFspq.X/

;

see [37, Proposition 2.3.2/2]. This finishes the proof of Theorem2.9.

Now we introduce the Besov,bBspq.X/, and the Triebel-Lizorkin,bFs

pq.X/, spaces.

DEFINITION 2.10. Let b be a para-accretive function as in Definition1.2, ž ∈.0; � ] and |s| < ž. Let {Sk}b

k∈Z be an approximation to the identity of orderž as inDefinition1.4andDk = Sk − Sk−1 for k ∈ Z.

(i) If max{d=.d + ž/; d=.d + s + ž/} < p ≤ ∞ and 0< q ≤ ∞, we define thespacebBs

pq.X/ to be the set of allf ∈ (Gb.þ; /)′

with

max{s; 0;−s + d.1=p − 1/+} < þ < ž and

max{s − d=p; d.1=p − 1/+;−s + d.1=p − 1/} < < ž(2.13)

such that

‖ f ‖bBspq.X/

={ ∞∑


L p.X/

}1=q

< ∞

with the usual modifications made whenp = ∞ or q = ∞.


(ii) If max{d=.d+ž/; d=.d+s+ž/} < p < ∞and max{d=.d+ž/; d=.d+s+ž/} <q ≤ ∞, we define the spacebFs

pq.X/ to be the set of allf ∈ (Gb.þ; /

)′with þ;

as in (2.13) such that

‖ f ‖bFspq.X/

=∥∥∥∥∥∥{ ∞∑

k=−∞2ksq|Dk. f /|q

}1=q∥∥∥∥∥∥

L p.X/

< ∞


Theorem2.2 and Theorem2.9 tell us that the definitions of the Besov spacebBs

pq.X/ and the Triebel-Lizorkin spacebFspq.X/ are independent of the choice of the

approximation to the identity and the distributional space,(Gb.þ; /

)′, with þ; as

in (2.13).

REMARK 2.11. To guarantee the definition of the spacesbBspq.X/ and bFs

pq.X/

is independent of the choice of the distribution space.Gb.þ; //′, we need only to

restrict that max{0;−s + d.1=p − 1/+} < þ < ž and max{0; s − d=p} < < ž;see Theorem2.9. However, if we restrict max.0; s/ < þ < ž and max{d.1=p −1/+;−s+ d.1=p− 1/} < < ž, we prove in the following proposition that the spaceof test functions,bGb.þ; /, is contained in the spacesbBs

pq.X/ andbFspq.X/. Thus,

the spacesbBspq.X/ andbFs

pq.X/ are non-empty if we restrictþ; as in (2.13).

PROPOSITION2.12. Letbbe a para-accretive function as in Definition1.2, ž ∈ .0; � ]and|s| < ž.

(i) If max{d=.d+ž/; d=.d+s+ž/}< p<∞ andmax{d=.d+ž/; d=.d+s+ž/} <q ≤ ∞, thenbBs

p;min.p;q/.X/ ⊂ bFspq.X/ ⊂ bBs

p;max.p;q/.X/.(ii) If f ∈ bGb.þ; / with max.0; s/ < þ < ž and max{d.1=p − 1/+;−s +

d.1=p−1/} < < ž, then f ∈ bBspq.X/withmax{d=.d+ž/; d=.d+s+ž/} < p ≤ ∞

and0 < q ≤ ∞, and f ∈ bFspq.X/ with max{d=.d + ž/; d=.d + s + ž/} < p < ∞

and max{d=.d + ž/; d=.d + s + ž/} < q ≤ ∞. Moreover, if p; q < ∞, then thespacebGb.þ; / with þ; as above is dense according to the norm‖ · ‖bBs

pq.X/in the

spacebBspq.X/, and according to the norm‖ · ‖bFs

pq.X/in the spacebFs

pq.X/.

(iii) If 1 < p < ∞, then bF0p2.X/ = L p.X/ with an equivalent norm; and if

d=.d + ž/ < p ≤ 1, thenbF0p2.X/ = H p.X/ with an equivalent norm.

PROOF. The proof of (i) is trivial; see [37, Proposition 2.3.2/2] and [40, Proposi-tion 2.3].

Let f ∈ bGb.þ; / with max.0; s/ < þ < ž and

max{d.1=p − 1/+;−s + d.1=p − 1/} < < ž:


Then f = bg for someg ∈ Gb.þ; / and‖ f ‖bGb.þ; / = ‖g‖Gb.þ; /. Let {Dk}k∈Z be thesame as in Definition2.10. The same argument as in (2.7) and (2.8) yields that fork = 0; 1; 2; : : : , − ∈ Ik and¹ = 1; 2; : : : ; N.k; − /,

|Dk. f /.x/| ≤ C 2−kþ‖ f ‖bGb.þ; /

1

.1 + ².y; x0//d+ ;(2.14)

and fork = −1;−2; : : : , − ∈ Ik and¹ = 1; 2; : : : ; N.k; − /,

|Dk. f /.x/| ≤ C 2k ′ ‖ f ‖bGb.þ; /

2−k

.2−k + ².y; x0//d+ ;(2.15)

where ′ can be any positive number in.0; /.From (2.14), (2.15) and Definition2.10, it follows that

‖ f ‖bBspq.X/

={ ∞∑


L p.X/

}1=q

≤ C‖ f ‖bGb.þ; /

{ ∞∑k=0

2k.s−þ/q +−1∑

k=−∞2k[s+ ′−d.1=p−1/]q

}1=q

≤ C‖ f ‖bGb.þ; /;

where we choose ′ ∈ .0; / such that ′ > −s + d.1=p − 1/. Thus, f ∈ bBspq.X/.

From this and (i), it is easy to deduce thatf is also inbFspq.X/.

The density of the spacebGb.þ; / in the spacebBspq.X/ and the spacebFs

pq.X/can be proved by the same argument as in [14, Proposition 3.8]. This proves (ii).

If 1 < p < ∞, David, Journe and Semmes in [6], proved thatbF0p2.X/ =

L p.X/ with an equivalent norm. Moreover, Han, Lee and Lin in [17], proved that ifd=.d + ž/ < p ≤ 1, thenbF0

p2.X/ = H p.X/ with an equivalent norm. This finishesthe proof of this proposition.

REMARK 2.13. Based on Proposition2.12 (iii), it is interesting to make clearunder what restrictions ons; p and q one will have thatbBs

pq.X/ = Bspq.X/ and

bFspq.X/ = Fs

pq.X/, with an equivalent norm.

Lemma2.6, Theorem2.5and the same argument as the proof of [16, Theorem 3]can also give the characterization of the Littlewood-PaleyS-function of the Triebel-Lizorkin spacebFs

pq.X/ as below. We omit the details.

THEOREM 2.14. Let b be a para-accretive function as in Definition1.2, ž ∈ .0; � ]and |s| < ž. Let {Sk}b

k∈Z be an approximation to the identity of orderž as in Defini-tion 1.4and Dk = Sk − Sk−1 for k ∈ Z. If max{d=.d + ž/; d=.d + s + ž/} < p < ∞


and max{d=.d + ž/; d=.d + s + ž/} < q ≤ ∞, then f ∈ bFspq.X/ if and only if

f ∈ .Gb.þ; //′ with þ; as in(2.13) and Ss

q. f / ∈ L p.X/, where

Ssq. f /.x/ =

{ ∞∑k=−∞

∫².x;y/≤2−k

2kd[2ks|Dk. f /.y/|]q

d¼.y/

}1=q

for all x ∈ X. Moreover, in this case,‖ f ‖bFspq.X/

∼ ‖Ssq. f /‖L p.X/.

We now establish the frame characterizations of the Besov spacebBspq.X/ and

the Triebel-Lizorkin spacebFspq.X/. To this end, we first introduce some spaces of

sequences,bspq.X/ and f s

pq.X/. Let

½ = {½k;¹− : k ∈ Z; − ∈ Ik; ¹ = 1; : : : ; N.k; − /

}(2.16)

be a sequence of complex numbers. The spacebspq.X/ with s ∈ R and 0< p;q ≤ ∞

is the set of all½ as in (2.16) such that

‖½‖bspq.X/

=

∞∑k=−∞

2ksq

[∑−∈Ik

N.k;− /∑¹=1

¼.Qk;¹− /

∣∣½k;¹−

∣∣p

]q=p

1=q

< ∞:

The spacef spq.X/ with s ∈ R, 0 < p < ∞, and 0< q ≤ ∞ is the set of all½ as in

(2.16) such that

‖½‖ f spq.X/

=∥∥∥∥∥∥{ ∞∑

k=−∞

∑−∈Ik

N.k;− /∑¹=1

2ksq∣∣½k;¹−

∣∣q �Qk;¹−

}1=q∥∥∥∥∥∥

L p.X/

< ∞:

THEOREM 2.15. Let b be a para-accretive function as in Definition1.2, ž ∈ .0; � ]and|s| < ž. With all the notation as in Lemma2.6, let½ be a sequence of numbers asin (2.16).

(i) If max{d=.d + ž/; d=.d + s+ ž/} < p ≤ ∞, 0< q ≤ ∞ and‖½‖bspq.X/

< ∞,then the series

∞∑k=−∞

∑−∈Ik

N.k;− /∑¹=1

½k;¹−

∫Qk;¹−

b.x/Ek.y; x/b.y/ d¼.y/(2.17)

converges to somef ∈ bBspq.X/ both in the norm ofbBs

pq.X/ and in.Gb.þ; //′ with

max{0;−s + d.1=p − 1/+} < þ < ž and max{0; s − d=p} < < ž(2.18)

whenp; q < ∞ and only in.Gb.þ; //′ with þ and as in(2.18) whenmax.p;q/ =

∞. Moreover, in all cases,‖ f ‖bBspq.X/

≤ C‖½‖bspq.X/

.


(ii) If max{d=.d+ž/; d=.d+s+ž/} < p < ∞, max{d=.d+ž/; d=.d+s+ž/} <q ≤ ∞ and‖½‖ f s

pq.X/< ∞, then the series in(2.17) converges to somef ∈ bFs

pq.X/

both in the norm ofbFspq.X/ and in.Gb.þ; //

′ with þ, as in(2.18) whenp;q < ∞and only in.Gb.þ; //

′ with þ and as in (2.18) whenmax.p;q/ = ∞. Moreover,in all cases,‖ f ‖bFs

pq.X/≤ C‖½‖ f s

pq.X/.

The proof of this theorem is similar to the proof of the frame characterizationsof the Besov spaceBs

pq.X/ and the Triebel-Lizorkin spaceFspq.X/ in [48]; see also

[23, 43]. We omit the details here.From Lemma2.6, Theorem2.14and the Plancherel-Polya inequalities, and Theo-

rem2.5, the frame characterizations of the spacesbBspq.X/ andbFs

pq.X/ follow.

THEOREM 2.16. Let b be a para-accretive function as in Definition1.2, ž ∈ .0; � ]and|s| < � . With all the notation as in Lemma2.6, let ½k;¹

− = Ek. f /.yk;¹− / for k ∈ Z,

− ∈ Ik and¹ = 1; : : : ; N.k; − /, whereyk;¹− is any fixed element inQk;¹

− .

(i) If max{d=.d + ž/; d=.d + s + ž/} < p ≤ ∞ and 0 < q ≤ ∞, thenf ∈ bBs

pq.X/ if and only if f ∈ .Gb.þ; //′ for someþ, as in (2.13), (2.2)

holds in.Gb.þ′; ′//′ with þ < þ ′ < ž and < ′ < ž, and½ ∈ bs

pq.X/. Moreover,in this case,‖ f ‖bBs

pq.X/∼ ‖½‖bs

pq.X/.

(ii) If max{d=.d + ž/; d=.d + s + ž/} < p < ∞ and max{d=.d + ž/; d=.d +s + ž/} < q ≤ ∞, then f ∈ bFs

pq.X/ if and only if f ∈ .Gb.þ; //′ for someþ,

as in (2.13), (2.2) holds in.Gb.þ′; ′//′ with þ < þ ′ < ž and < ′ < ž, and

½ ∈ f spq.X/. Moreover, in this case,‖ f ‖bFs

pq.X/∼ ‖½‖ f s

pq.X/.

3. Spacesb−1Bspq(X) and b−1F s

pq(X)

In this section, we introduce another new Besov spaceb−1Bspq.X/ and new Triebel-

Lizorkin spaceb−1Fspq.X/, which are closely related to the Besov spacebBs

pq.X/ andthe Triebel-Lizorkin spacebFs

pq.X/, respectively.

DEFINITION 3.1. Let b be a para-accretive function as in Definition1.2, ž ∈ .0; � ]and|s| < ž. Let {Sk}b

k∈Z be an approximation to the identity of orderž as in Defini-tion 1.4andDk = Sk − Sk−1 for k ∈ Z.

(i) If max{d=.d + ž/; d=.d + s + ž/} < p ≤ ∞ and 0< q ≤ ∞, we define thespaceb−1Bs

pq.X/ to be the set of allf ∈ .bGb.þ; //′ with þ, as in (2.13) such that

‖ f ‖b−1 Bspq.X/

={ ∞∑

k=−∞2ksq‖Dk.bf /‖q

L p.X/

}1=q

< ∞

with the usual modifications made whenp = ∞ or q = ∞.


(ii) If max{d=.d+ž/; d=.d+s+ž/} < p < ∞and max{d=.d+ž/; d=.d+s+ž/} <q ≤ ∞, we define the spaceb−1Fs

pq.X/ to be the set of allf ∈ .bGb.þ; //′ with þ,

as in (2.13) such that

‖ f ‖b−1 Fspq.X/

=∥∥∥∥∥∥{ ∞∑

k=−∞2ksq|Dk.bf /|q

}1=q∥∥∥∥∥∥

L p.X/

< ∞


To verify that Definition3.1 is independent of the choice of the approximation tothe identity, we first need to establish the following Plancherel-Polya inequality.




(i) If max{d=.d + ž/; d=.d + s + ž/} 0 such that for all f ∈ .bGb.þ; //

′ with 0< þ; < ž, we have

∞∑k=−∞

2ksq

[∑−∈Ik

N.k;− /∑¹=1

¼.Qk;¹− / sup

z∈Qk;¹−

|Dk.bf /.z/|p

]q=p

1=q

≤ C

∞∑k=−∞

2ksq

[∑−∈Ik

N.k;− /∑¹=1

¼.Qk;¹− / inf

z∈Qk;¹−

|Ek.bf /.z/|p

]q=p

1=q

:

(ii) If max{d=.d + ž/; d=.d + s + ž/} 0 such that for all f ∈ .bGb.þ; //

′ with0< þ; < ž, we have∥∥∥∥∥∥

{ ∞∑k=−∞

∑−∈Ik

N.k;− /∑¹=1

2ksq supz∈Qk;¹

−

|Dk.bf /.z/|q�Qk;¹−

}1=q∥∥∥∥∥∥

L p.X/

≤ C

∥∥∥∥∥∥{ ∞∑

k=−∞

∑−∈Ik

N.k;− /∑¹=1

2ksq infz∈Qk;¹

−

|Ek.bf /.z/|q�Qk;¹−

}1=q∥∥∥∥∥∥

L p.X/

:

Theorem3.2 can be proved in a way similar to Theorem2.5, if we replaceLemma2.6 by the following discrete Calderon reproducing formula in [17]. Weomit the details.

LEMMA 3.3. Letb be a para-accretive function as in Definition1.2. Letž ∈ .0; � ],{Gk}b

k∈Z be an approximation to the identity of orderž as in Definition1.4, and


Ek = Gk − Gk−1 for k ∈ Z. Let0< þ; < ž. Then there exists a family of functions{Ek.x; y/}k∈Z such that for all fixedyk;¹

− ∈ Qk;¹− and all f ∈ .bGb.þ; //

′,

f .x/ =∑k∈Z

∑−∈Ik

N.k;− /∑¹=1

Ek.bf /.yk;¹− /

∫Qk;¹−

Ek.y; x/b.y/ d¼.y/;(3.1)

where Ek.x; y/ for k ∈ Z satisfies(i) and (iii) of Definition1.4 with ž replaced byž′ ∈ .0; ž/, and∫

X

Ek.x; y/b.x/ d¼.x/ =∫

X

Ek.x; y/b.y/ d¼.y/ = 0:

Moreover the series in(3.1) converges in the sense that for allg ∈ bGb.þ′; ′/ with

þ < þ ′ < ž and < ′ < ž,

limM; N→∞

⟨∑|k|≤M

∑−∈Ik

².x0;zk− /≤N

N.k;− /∑¹=1

Ek.bf /.yk;¹− /

∫Qk;¹−

Ek.y; ·/b.y/ d¼.y/; g

⟩= 〈 f; g〉:

By Lemma3.3and Theorem3.2, we can also obtain a counterpart of Theorem2.9by the same procedure as in Theorem2.9. Thus, Definition3.1is also independent ofthe choice of the distribution space.bGb.þ; //

′ with þ, as in (2.13); and, therefore,Definition3.1 is reasonable.

REMARK 3.4. To guarantee the definition of the spacesb−1Bspq.X/ andb−1Fs

pq.X/

is independent of the choice of the distribution space.bGb.þ; //′, we need only to

restrict that max{0;−s + d.1=p − 1/+} < þ < ž and max{0; s − d=p} < < ž;see the proof of Theorem2.9. However, if we restrict max.0; s/ < þ < ž andmax{d.1=p − 1/+;−s + d.1=p − 1/} < < ž, we can prove that the space of testfunctions,Gb.þ; /, is contained in the spacesb−1Bs

pq.X/ andb−1Fspq.X/ in a way

similar to that of Proposition2.12. Thus, the spacesb−1Bspq.X/ andb−1Fs

pq.X/ arenon-empty if we restrictþ, as in (2.13).

Moreover, using Lemma3.3and similar to proofs of Theorems2.15and2.16, wecan also obtain the frame characterizations of the spacesb−1Bs

pq.X/ andb−1Fspq.X/.

We will not give the details of these proofs.

THEOREM 3.5. Let b be a para-accretive function as in Definition1.2, ž ∈ .0; � ]and|s| < ž. Let all the notation be the same as in Lemma3.3and½ be a sequence ofnumbers as in(2.16).


(i) If max{d=.d + ž/; d=.d + s+ ž/} < p ≤ ∞, 0< q ≤ ∞ and‖½‖bspq.X/

< ∞,then the series

∞∑k=−∞

∑−∈Ik

N.k;− /∑¹=1

½k;¹−

∫Qk;¹−

Ek.y; x/b.y/ d¼.y/(3.2)

converges to somef ∈ b−1Bspq.X/ both in the norm ofb−1Bs

pq.X/ and in.bGb.þ; //′

with

max{0;−s + d.1=p − 1/+} < þ < ž and max{0; s − d=p} < < ž(3.3)

whenp;q < ∞ and only in.bGb.þ; //′ with þ and as in(3.3) whenmax.p;q/ =

∞. Moreover, in all cases,‖ f ‖b−1 Bspq.X/

≤ C‖½‖bspq.X/

.(ii) If max{d=.d+ž/; d=.d+s+ž/} < p < ∞, max{d=.d+ž/; d=.d+s+ž/} <

q ≤ ∞, and‖½‖ f spq.X/

< ∞, then the series in(3.2) converges to somef ∈ b−1Fspq.X/

both in the norm ofb−1 Fspq.X/and in.bGb.þ; //

′ withþ; as in(3.3) whenp;q < ∞and only in.bGb.þ; //

′ with þ and as in (3.3) whenmax.p;q/ = ∞. Moreover,in all cases,‖ f ‖b−1 Fs

pq.X/≤ C‖½‖ f s

pq.X/.

THEOREM 3.6. Let b be a para-accretive function as in Definition1.2, ž ∈ .0; � ]and|s| < � . With all the notation as in Lemma3.3, let½k;¹

− = Ek.bf /.yk;¹− / for k ∈ Z,

− ∈ Ik and¹ = 1; : : : ; N.k; − /, whereyk;¹− is any fixed element inQk;¹

− .

(i) If max{d=.d + ž/; d=.d + s + ž/} < p ≤ ∞ and 0 < q ≤ ∞, then f ∈b−1Bs

pq.X/ if and only if f ∈ .bGb.þ; //′ for someþ, as in (2.13), (3.1) holds in

.bGb.þ′; ′//′ with þ < þ ′ < ž and < ′ < ž, and½ ∈ bs

pq.X/. Moreover, in thiscase,‖ f ‖b−1 Bs

pq.X/∼ ‖½‖bs

pq.X/.

(ii) If max{d=.d + ž/; d=.d + s + ž/} < p < ∞ and max{d=.d + ž/; d=.d +s + ž/} < q ≤ ∞, then f ∈ b−1Fs

pq.X/ if and only if f ∈ .bGb.þ; //′ for some

þ, as in(2.13), (3.1) holds in.bGb.þ′; ′//′ with þ < þ ′ < ž and < ′ < ž, and

½ ∈ f spq.X/. Moreover, in this case,‖ f ‖b−1 Fs

pq.X/∼ ‖½‖ f s

pq.X/.

We also have the characterization of the Littlewood-PaleyS-function of the Triebel-Lizorkin spaceb−1Fs

pq.X/, which can be proved in a way similar to Theorem2.14.

THEOREM3.7. Letb be a para-accretive function as in Definition1.2, ž ∈ .0; � ] and|s| < ž. Let{Sk}b

k∈Z be an approximation to the identity of orderž as in Definition1.4and Dk = Sk − Sk−1 for k ∈ Z. If max{d=.d + ž/; d=.d + s + ž/} < p < ∞andmax{d=.d + ž/; d=.d + s + ž/} < q ≤ ∞, then f ∈ b−1Fs

pq.X/ if and only if

f ∈ .bGb.þ; //′ with þ, as in(2.13) and Ss

q;b. f / ∈ L p.X/, where

Ssq;b. f /.x/ =

{ ∞∑k=−∞

∫².x;y/≤2−k

2kd[2ks|Dk.bf /.y/|]q

d¼.y/

}1=q


for all x ∈ X. Moreover, in this case,‖ f ‖b−1 Fspq.X/

∼ ‖Ssq;b. f /‖L p.X/.

From Definition 2.10, Definition 3.1 and (1.3), it is easy to deduce the belowrelations between the spacebBs

pq.X/ and the spaceb−1Bspq.X/, and between the space

bFspq.X/ and the spaceb−1Fs

pq.X/.On the other hand, ford=.d + ž/ < p ≤ 1, let H p.X/ be the Hardy spaces studied

by Macıas and Segovia in [28]. We define the Hardy spaceH pb .X/ to be the set of all

f ∈ .bGb.þ; //′ with d.1=p − 1/+ < þ; < ž such thatbf ∈ H p.X/. Moreover,

we define‖ f ‖H pb .X/ = ‖bf ‖H p.X/. When X = Rn, H1

b .X/ was first introduced byMeyer in [31] and H p

b .X/ was first introduced by Han, Lee and Lin in [17]. Basedon the results in [17, 16], we can easily obtain the relation between the Hardy spacesH p

b .X/ and the Triebel-Lizorkin spacesb−1Fspq.X/.

PROPOSITION3.8. Letb be a para-accretive function as in Definition1.2, ž ∈ .0; � ]and|s| < ž.

(i) If max{d=.d + ž/; d=.d + s + ž/} < p ≤ ∞ and 0 < q ≤ ∞, then f ∈b−1Bs

pq.X/ if and only ifbf ∈ bBspq.X/. Moreover, in this case,


= ‖bf ‖bBspq.X/

:

(ii) If max{d=.d+ž/; d=.d+s+ž/} < p < ∞andmax{d=.d+ž/; d=.d+s+ž/} <q ≤ ∞, then f ∈ b−1Fs

pq.X/ if and only ifbf ∈ bFspq.X/. Moreover, in this case,


= ‖bf ‖bBspq.X/

:

(iii) If 1 < p < ∞, thenb−1F0p2.X/ = L p.X/ with an equivalent norm; and if

d=.d + ž/ < p ≤ 1, thenb−1F0p2.X/ = H p

b .X/ with an equivalent norm.

4. Some applications

We first consider real interpolations of the spacesbBspq.X/, bFs

pq.X/, b−1Bspq.X/,

andb−1Fspq.X/. Let us now recall the general background of the real interpolation

method; see [1] and [38, pages 62–64].LetH be a linear complex Hausdorff space, and letA0 andA1 be two complex

quasi-Banach spaces such thatA0 ⊂H andA1 ⊂H . LetA0 +A1 be the set of allelementsa ∈H which can be represented asa = a0 + a1 with a0 ∈ A0 anda1 ∈ A1.If 0 < t < ∞ anda ∈ A0 +A1, then Peetre’s celebratedK -functional is given by

K .t; a/ = K .t; a;A0;A1/ = inf(‖a0‖A0 + t‖a1‖A1

);

where the infimum is taken over all representations ofa having the forma = a0 + a1

with a0 ∈ A0 anda1 ∈ A1.


DEFINITION 4.1. Let 0< ¦ < 1. If 0 < q < ∞, then

.A0;A1/¦;q ={

a : a ∈ A0 +A1; ‖a‖.A0;A1/¦;q ={∫ ∞

0

[t−¦ K .t; a/

]q dt

t

}1=q

< ∞}:

If q = ∞, then

.A0;A1/¦;∞ ={

a : a ∈ A0 +A1; ‖a‖.A0;A1/¦;∞ = sup0<t<∞

t−¦ K .t; a/ < ∞}:

Using Lemma2.6, Theorems2.15–2.16, Lemma3.3, Theorems3.5–3.6, and themethod of retraction and coretraction as in the proofs of [37, Theorem 2.4.1 andTheorem 2.4.2], we obtain theorems on the real interpolations of the spacesbBs

pq.X/,bFs

pq.X/, b−1Bspq.X/ andb−1Fs

pq.X/. We omit the details; see also the proofs of [43,Theorem 3.2 and Theorem 3.3].

THEOREM 4.2. Let b be a para-accretive function as in Definition1.2, ž ∈ .0; � ]and¦ ∈ .0; 1/.

(i) Let−ž < s0; s1 < ž, s0 6= s1, 1 ≤ p ≤ ∞, and0< q0;q1;q ≤ ∞. Then(bBs0

p;q0.X/; bBs1

p;q1.X/

)¦;q

= bBspq.X/

and (b−1Bs0

p;q0.X/; b−1Bs1

p;q1.X/

)¦;q

= b−1Bspq.X/;

wheres = .1 − ¦/s0 + ¦s1.(ii) Let−ž < s< ž, 1 ≤ p ≤ ∞, 0< q0;q1 ≤ ∞, andq0 6= q1. Then(

bBsp;q0.X/; bBs

p;q1.X/

)¦;q

= bBspq.X/

and (b−1Bs

p;q0.X/; b−1Bs

p;q1.X/

)¦;q

= b−1Bspq.X/;

where1=q = .1 − ¦/=q0 + ¦=q1.(iii) Let−ž < s0; s1 < ž and1 ≤ p0; p1 ≤ ∞. Then(

bBs0p0;p0

.X/; bBs1p1;p1

.X/)¦;p

= bBsp;p.X/

and (b−1Bs0

p0;p0.X/; b−1Bs1

p1;p1.X/

)¦;p

= b−1Bsp;p.X/;

where1=p = .1 − ¦/=p0 + ¦=p1.

THEOREM 4.3. Let b be a para-accretive function as in Definition1.2, ž ∈ .0; � ]and¦ ∈ .0; 1/. Let−ž < s0; s1 < ž, max.d=.d + ž/; d=.d + ž + s0// < p0 < ∞,max.d=.d + ž/; d=.d + ž + s1// < p1 < ∞, 1 ≤ q0;q1 ≤ ∞, s = .1 − ¦/s0 + ¦s1,1=p = .1 − ¦/=p0 + ¦=p1, and1=q = .1 − ¦/=q0 + ¦=q1.


(i) If s0 6= s1, then(bFs0

p0;q0.X/; bFs1

p1;q1.X/

)¦;p

= bBsp;p.X/

( = bFsp;p.X/

)and (

b−1Fs0p0;q0

.X/; b−1Fs1p1;q1

.X/)¦;p

= b−1Bsp;p.X/

( = b−1Fsp;p.X/

):

(ii) If s0 = s1 = s, p0 = q0, p1 = q1, andq0 6= q1, then(bFs

p0;p0.X/; bFs

p1;p1.X/

)¦;p

= bBsp;p.X/

and (b−1Fs

p0;p0.X/; b−1Fs

p1;p1.X/

)¦;p

= b−1Bsp;p.X/:

(iii) If s0 = s1 = s, q0 = q1 = q, and p0 6= p1, then(bFs

p0;q.X/; bFs

p1;q.X/

)¦;p

= bFspq.X/

and (b−1Fs

p0;q.X/; b−1Fs

p1;q.X/

)¦;p

= b−1Fspq.X/:

Moreover, by Lemma2.6, Lemma3.3, and some similar computations to the proofof [47, Theorem 2.1], we can further establish the following general interpolationtheorem; see also [37, Theorem 2.4.2].

THEOREM 4.4. Let b be a para-accretive function as in Definition1.2, ž ∈ .0; � ],¦ ∈ .0; 1/, s0; s1 ∈ .−ž; ž/, s0 6= s1, ands = .1 − ¦/s0 + ¦s1.

(i) If max.d=.d + ž/; d=.d + s0 + ž/; d=.d + s1 + ž// < p ≤ ∞ and 0 <

q0;q1;q ≤ ∞, then (bBs0

p;q0.X/; bBs1

p;q1.X/

)¦;q

= bBspq.X/

and (b−1Bs0

p;q0.X/; b−1Bs1

p;q1.X/

)¦;q

= b−1Bspq.X/:

(ii) If max.d=.d + ž/; d=.d + s0 + ž/; d=.d + s1 + ž// < p < ∞, max.d=.d +ž/; d=.d + si + ž// < qi ≤ ∞ for i = 0; 1, and0< q ≤ ∞, then(

bFs0p;q0.X/; bFs1

p;q1.X/

)¦;q

= bBspq.X/

and (b−1Fs0

p;q0.X/; b−1Fs1

p;q1.X/

)¦;q

= b−1Bspq.X/:

By Lemma2.6and Lemma3.3with the estimate (2.4), and the same argument asin [18] (see also [15, 23, 44]), we can also obtain the following embedding theorem.In the sequel, for two quasi-Banach spacesA1 andA2, A1 ⊂ A2 means a linear andcontinuous embedding.


THEOREM 4.5. Let b be a para-accretive function as in Definition1.2, ž ∈ .0; � ]and−ž < s2 < s1 < ž.

(i) If 0 < q ≤ ∞, max.d=.d + ž/; d=.d + ž + si // < pi ≤ ∞ for i = 1; 2, ands1−d=p1 = s2 −d=p2, thenbBs1

p1;q.X/ ⊂ bBs2

p2;q.X/ andb−1Bs1

p1;q.X/ ⊂ b−1Bs2

p2;q.X/;

(ii) If max.d=.d + ž/; d=.d+ ž+si // < pi < ∞ andmax.d=.d+ ž/; d=.d + ž+si // < qi ≤ ∞ for i = 1; 2, ands1 −d=p1 = s2 −d=p2, thenbFs1

p1;q1.X/ ⊂ bFs2

p2;q2.X/

andb−1Fs1p1;q1

.X/ ⊂ b−1Fs2p2;q2

.X/.

REMARK 4.6. In [15, 18, 23, 44], all the results similar to Theorem4.5asked that−ž < s1 − d=p1 = s2 − d=p2 < ž. However,s2 − d=p2 < ž is automatically truesinces2 < ž. A careful check of their proofs shows that one has not used the condition−ž < s1 − d=p1.

We now turn to consider theT b theorems on the spacesbBspq.X/, bFs

pq.X/,b−1Bs

pq.X/ and b−1Fspq.X/. We first recall some notation. In what follows, for

� ∈ .0; � ], we letC�

0.X/ be the set of all functions having compact support such that

‖ f ‖C�

0 .X/ = supx 6=y

| f .x/− f .y/|².x; y/�

< ∞:

EndowC�

0.X/with the natural topology and let.C�

0.X//′ be its dual space. Moreover,

if b is a para-accretive function as in Definition1.2, in what follows, we will useMb

to denote the corresponding multiplication operator andbC�

0.X/ to denote the imageof C�

0.X/ underMb with the natural topology. This means thatf ∈ bC�

0.X/ if andonly if f = bg for someg ∈ C�

0.X/ and we define‖ f ‖bC�0 .X/ = ‖g‖C�

0.X/.Let b1 andb2 be two para-accretive functions as in Definition1.2. A continuous

complex-valued functionK .x; y/ on � = {.x; y/ ∈ X × X : x 6= y} is called aCalderon-Zygmund kernel of typež if there existž ∈ .0; � ] andC6 > 0 such that for².x; y/ 6= 0,

|K .x; y/| ≤ C6².x; y/−d;(4.1)

and for².x; x′/ ≤ ².x; y/=.2A/,

|K .x; y/− K .x′; y/| ≤ C6².x; x′/ž².x; y/−d−ž;(4.2)

and for².y; y′/ ≤ ².x; y/=.2A/,

|K .x; y/− K .x; y′/| ≤ C6².y; y′/ž².x; y/−d−ž;(4.3)

and a continuous linear operatorT : b1C�

0.X/ → .b2C�

0.X//′ is aCalderon-Zygmund

singular integral operator of typež if there is a Calderon-Zygmund kernelK .x; y/ oftypež such that

〈T f; g〉 =∫

X

∫X

g.x/b2.x/K .x; y/b1.y/ f .y/ d¼.x/ d¼.y/


for all f; g ∈ C�

0.X/ with disjoint supports. Moreover, a Calderon-Zygmund singularintegral operatorT is said to have the weak boundedness property, if there exist� ∈ .0; � ] andC7 > 0 such that

|〈T f; g〉| ≤ C7rd+2�‖ f ‖C�

0 .X/‖g‖C�

0 .X/

for all f; g ∈ C�

0.X/ with diam.supp f / ≤ r and diam.suppg/ ≤ r , and we denotethis byT ∈ W B P.X/.

THEOREM 4.7. Let b be a para-accretive function as in Definition1.2, ž ∈ .0; � ],and|s| < ž. SupposeT is a Calderon-Zygmund singular integral operator of typež,T.b/ = 0 = T ∗.b/, MbT Mb ∈ W B P, and its kernelK .x; y/ satisfies(4.1), (4.2),and (4.3).

(i) If max.d=.d+ž/; d=.d+s+ž// < p ≤ ∞ and0< q ≤ ∞, thenT is boundedfrom bBs

pq.X/ to b−1Bspq.X/ with an operator norm not larger thanC max.C6;C7/;

(ii) If max.d=.d + ž/; d=.d + s+ ž// < p < ∞ andmax.d=.d + ž/; d=.d + s+ž// < q ≤ ∞, thenT is bounded frombFs

pq.X/ to b−1Fspq.X/ with an operator norm

not larger thanC max.C6;C7/.

PROOF. We only give an outline of the proof. Let{D j } j ∈Z be as in Definition2.10.With all the notation as in Lemma2.6, under the assumptions of the theorem, we canverify that for all j ; k ∈ Z and allx; y ∈ X,

∣∣∣[Dj MbT MbEk.y; ·/].x/∣∣∣ ≤ C2−|k− j |ž′ 2−.k∧ j /ž′

.2−.k∧ j / + ².x; y//d+ž′ ;(4.4)

where ž′ can be any positive number in.0; ž/; see [14, Lemma 3.13] and [46,Lemma 2.3] for details. From the estimate (4.4), Lemma2.6and Lemma2.7, and byan argument similar to the proof of Theorem2.5, we can prove (ii).

The estimate (4.4) and some trivial computation also lead to the conclusion (i) withp = q = ∞. This, together with (ii) and Theorem4.4, will then give (i); see also [46]for details. This completes the proof of Theorem4.7.

Finally, we consider the boundedness of Riesz potentials on the spacesbBspq.X/,

bFspq.X/, b−1Bs

pq.X/, andb−1Fspq.X/.

DEFINITION 4.8. Let b be a para-accretive function as in Definition1.2, ž ∈ .0; � ],{Dk}k∈Z be the same as in Definition2.10, andÞ ∈ R. Then the Riesz operatorIÞ forf ∈ Gb.þ; / with 0 < þ; < ž is defined byIÞ. f /.x/ = ∑∞

l=−∞ Dl . f /.x/ for allx ∈ X.


Obviously, whenÞ > 0 andb ≡ 1, IÞ is the discrete version of the fractionalintegrals introduced in [10, 11, 9]; while whenÞ < 0 andb ≡ 1, IÞ is the discreteversion of the fractional derivatives introduced there. WhenÞ = 0 andb ≡ 1, IÞ isjust the identity. We also mention that in [32, 33], Nahmod considered some discreteand inhomogeneous fractional integrals and derivatives similar to those above.

THEOREM 4.9. Let b be a para-accretive function as in Definition1.2, ž ∈ .0; � ],|Þ| < ž, |s| < ž, and|s + Þ| < ž.

(i) If max{d=.d + ž/; d=.d + ž + s + Þ ∧ 0/; d=.d + ž − Þ/} < p ≤ ∞, and0< q ≤ ∞, thenIÞ is bounded frombBs

pq.X/ into b−1Bs+Þpq .X/.

(ii) If max{d=.d + ž/; d=.d + ž + s + Þ ∧ 0/; d=.d + ž − Þ/} < p < ∞, andmax{d=.d + ž/; d=.d + ž+s+Þ∧0/; d=.d + ž−Þ//} < q ≤ ∞, thenIÞ is boundedfrom bFs

pq.X/ into b−1Fs+Þpq .X/.

PROOF. We only give an outline; see [45, 46] for details. Let{Dk}k∈Z be as inDefinition 2.10. With all the notation as in Lemma2.6, under the assumptions of thetheorem, we can verify that for allk; k′ ∈ Z and allx; y ∈ X,∣∣∣[Dk MbIÞMbEk′.y; ·/

].x/∣∣∣(4.5)

≤ C 2−.k∧k′/Þ2−|k−k′|.ž′+Þ∧0/ 2−.k∧k′/.ž′−Þ/

.2−.k∧k′/ + ².x; y//d+ž′−Þ ;

wherež′ can be any positive number in.0; ž/; see [45, Lemma 2] for details. Theestimate (4.5), Lemma2.6, and an argument similar to the proof of Theorem2.5yield (ii).

From the estimate (4.5) and some trivial computation, the conclusion (i) withp = q = ∞ can be deduced. This, together with (ii) and Theorem4.4 will thengive (i); see also [45] for details. This completes the proof of Theorem4.9.

REMARK 4.10. Theorem 1 and Theorem 2 in [45] also ask thats< � +Þ∧0. Thisis superfluous and the mistake is caused by the factor 2−kÞ in (2.1) there, which shouldbe 2−.k∧k′/Þ as in (4.5).

From Theorem4.5 and Theorem4.9, we can deduce the following interestingconclusion.

COROLLARY 4.11. Letb be a para-accretive function as in Definition1.2, ž ∈ .0; � ],0 ≤ Þ < ž and|s| < ž.

(i) If max{d=.d + ž/; d=.d + ž + s/; d=.d + ž − Þ/} < p1 ≤ ∞, 0 < q ≤ ∞,and1=p2 = 1=p1 − Þ=d, thenIÞ is bounded frombBs

p1;q.X/ into b−1Bs

p2;q.X/.


(ii) If max{d=.d + ž/; d=.d + ž + s/; d=.d + ž − Þ/} < p1 < ∞, max{d=.d +ž/; d=.d + ž + s/; d=.d + ž − Þ/} < q ≤ ∞, and1=p2 = 1=p1 − Þ=d, then IÞ isbounded frombFs

p1;q.X/ into b−1Fs

p2;q.X/.

We remark that Corollary4.11 (ii) is specially interesting by noting Proposi-tion2.12(iii) and Proposition3.8(iii). It means thatIÞ is bounded from the Hardy spaceH p1.X/ into the Hardy spaceH p2

b .X/, wherep1 and p2 are as in Corollary4.11(ii).Using Theorem4.7, we can also establish the converse of Theorem4.9.

THEOREM4.12. Let b be a para-accretive function as in Definition1.2, ž ∈ .0; � ],|Þ| < ž, |s| < ž and|s + Þ| < ž.

(i) If max{d=.d + ž/; d=.d + ž + Þ/; d=.d + ž + s − |Þ|/} 0 such that if|Þ| < Þ0.s/, for all f ∈ bBs

pq.X/, ‖ f ‖bBspq.X/

≤ C‖IÞ. f /‖b−1 Bs+Þpq .X/.

(ii) If max{d=.d + ž/; d=.d + ž + Þ/; d=.d + ž + s − |Þ|/} 0 such that if|Þ| < Þ0.s/, for all f ∈ bFs

pq.X/,‖ f ‖bFs

pq.X/≤ C‖IÞ. f /‖b−1 Fs+Þ

pq .X/.

PROOF. We only give an outline of the proof. The key of the proof is to verifythat the operatorI−ÞMbIÞMb is invertible in the spacesb−1Bs

pq.X/ andb−1Fspq.X/,

respectively. To this end, we need to show that the operatorT = I − I−ÞMbIÞMb isbounded on the spacesb−1Bs

pq.X/ andb−1Fspq.X/ with an operator norm less than 1

whenÞ is small. We show this by using Theorem4.7. In fact, by using Coifman’sidea in [6], for any givenN ∈ N, we write

T = I − I−ÞMbIÞMb =∞∑

k=−∞

∞∑l=−∞

.1 − 2−lÞ/Dk MbDk+l Mb

={ ∞∑

k=−∞

∑|l |≤N

.1−2−lÞ/Dk MbDk+l +∞∑

k=−∞

∑|l |>N

.1−2−lÞ/Dk MbDk+l

}Mb = T Mb:

It is easy to see thatT.b/ = 0 = T∗.b/. For anyž ′ ∈ .0; ž/, all k; l ∈ Z and allx; y ∈ X, recall

|Dk MbDk+l .x; y/| ≤ C 2−|l |ž′ 2−[k∧.k+l /]ž

.2−[k∧.k+l /] + ².x; y//d+ž ;(4.6)

and if².y; y′/ ≤ .1=4A2/².x; y/, for all ¦ ∈ .0; 1/,∣∣Dk MbDk+l .x; y/− Dk MbDk+l .x; y′/∣∣(4.7)

+ ∣∣Dk MbDk+l .y; x/− Dk MbDk+l .y′; x/

∣∣


≤ C 2−|l |ž′¦(

².y; y′/2−[k∧.k+l /] + ².x; y/

).1−¦/ž 2−[k∧.k+l /]ž

.2−[k∧.k+l /] + ².x; y//d+ž ;

see [14] for details.From the estimates (4.6) and (4.7), we can verify thatT is a Calderon-Zygmund

singular integral operator of type.1 − ¦/ž with

C6;C7 ≤ C8

∑|l |≤N

∣∣1 − 2−lÞ∣∣2−|l |¦ž ′ + C92

−ŽN;

whereŽ = min{¦ž + Þ; ¦ž − Þ}, C8 is independent ofÞ and N, C9 is independentof N, and if |Þ| < Þ1, whereÞ1 > 0 (and its value will be chosen later), thenC9 isalso independent ofÞ, but it may depend onÞ1.

By Theorem4.7, we know thatT is bounded frombBspq.X/ to b−1Bs

pq.X/ and frombFs

pq.X/ to b−1Fspq.X/ with an operator norm no more thanC10 = C max{C6;C7}.

Note thatMb is bounded fromb−1 Bspq.X/ tobBs

pq.X/, and fromb−1 Fspq.X/ tobFs

pq.X/with an operator norm to be 1. Thus,T is bounded onb−1Bs

pq.X/ andb−1Fspq.X/

with an operator norm no more thanC10. Now, if we chooseÞ1 small enough andif |Þ| < Þ1, then I−ÞMbIÞMb is invertible in the spacesb−1Bs

pq.X/ andb−1Fspq.X/.

Thus, by Theorem4.9, we have that for allf ∈ b−1Bspq.X/,


= ∥∥.I−ÞMbIÞMb/−1 I−ÞMbIÞMb. f /

∥∥b−1 Bs

pq.X/

≤ C ‖I−ÞMbIÞMb. f /‖b−1 Bspq.X/

≤ C ‖MbIÞMb. f /‖bBs+Þpq .X/

≤ C ‖IÞMb. f /‖b−1 Bs+Þpq .X/ ;

and for all f ∈ b−1Fspq.X/,

‖ f ‖b−1 Fspq.X/

= ∥∥.I−ÞMbIÞMb/−1 I−ÞMbIÞMb. f /

∥∥b−1 Fs

pq.X/

≤ C ‖I−ÞMbIÞMb. f /‖b−1 Fspq.X/

≤ C ‖MbIÞMb. f /‖bFs+Þpq .X/

≤ C ‖IÞMb. f /‖b−1 Fs+Þpq .X/ :

Proposition3.8then tells us the conclusion of the theorem.

By combining Theorem4.9 with Theorem4.12, we obtain the following simpleconclusion.

COROLLARY 4.13. Let |Þ| < ž, |s| < ž, and|s + Þ| < ž.


(i) If max{d=.d + ž/; d=.d + ž − |Þ|/; d=.d + ž + s − |Þ|/} < p ≤ ∞ and0 < q ≤ ∞, then there is anÞ0.s/ ∈ .0; ž/ such that if |Þ| < Þ0.s/, for allf ∈ bBs

pq.X/, ‖ f ‖bBspq.X/

∼ ‖IÞ. f /‖b−1 Bs+Þpq .X/.

(ii) If max{d=.d + ž/; d=.d + ž − |Þ|/; d=.d + ž + s − |Þ|/} < p < ∞ andmax{d=.d+ž/; d=.d+ž+Þ/; d=.d+ž+s−|Þ|/} < q ≤ ∞, then there is anÞ0.s/ ∈.0; ž/ such that if|Þ| < Þ0.s/, for all f ∈ bFs

pq.X/, ‖ f ‖bFspq.X/

∼ ‖IÞ. f /‖b−1 Fs+Þpq .X/.

From Corollary4.13, it is easy to see thatIÞ can be used as a lifting operatorfor the spacesbBs

pq.X/ and bFspq.X/ whenÞ is small; see also [38] for Rn case.

Moreover, by Corollary4.13, we can also see thatIÞ is independent of the choice ofthe approximation to the identity; see also [23, 45].

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Department of MathematicsZhongshan UniversityGuangzhou 510275People’s Republic of Chinae-mail: [email protected]

School of Mathematical SciencesBeijing Normal University

Beijing 100875People’s Republic of China

e-mail: [email protected]

mailto:[email protected]

mailto:[email protected]

Documents

· J. Aust. Math. Soc. 80 (2006), 229–262 SOME NEW BESOV AND TRIEBEL-LIZORKIN SPACES ASSOCIATED WITH PARA-ACCRETIVE FUNCTIONS ON SPACES OF HOMOGENEOUS TYPE DONGGUO DENG and DACHUN