# Herz-Type Hardy Spaces Associated with Operators

Authors: Yan Chai; Yaoyao Han; Kai Zhao
Journal: Journal of Function Spaces
 (2018)

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## Abstract

SupposeL is a nonnegative, self-adjoint differential operator. In this paper, we introduce the Herz-type Hardy spaces associated with operator L. Then, similar to the atomic and molecular decompositions of classical Herz-type Hardy spaces and the Hardy space associated with operators, we prove the atomic and molecular decompositions of the Herz-type Hardy spaces associated with operator L. As applications, the boundedness of some singular integral operators on Herz-type Hardy spaces associated with operators is obtained.


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## Body

## 1. Introduction

As we know, the theory of function spaces constitutes an important part of harmonic analysis and partial differential equations. Some results of the classical Hardy spaces can be found in [1–6], etc. Since there are some important situations in which the theory of classical Hardy spaces is not applicable, many authors begin to study Hardy spaces that are adapted to the differential operator L. For example, Auscher, Duong, and McIntosh [7], then Duong and Yan [8, 9], introduced the Hardy and BMO spaces adapted to the operator L which satisfies the Gaussian heat kernel upper bounds. Yang and his cooperators discussed new Orlicz-Hardy spaces associated with operators [10–13]. For more results, we refer to [14–19] and the references therein.It is known that many classical function spaces and the Hardy type spaces associated with operators have the atomic decompositions and the molecular decompositions, and the atomic and molecular decompositions of function spaces make the linear operators acting on spaces very simple; see [20–29], etc. In fact, the characterizations of spaces of functions or distributions, including the atomic and molecular characterizations, have many important applications in harmonic analysis. In recent years, it has been proved that many results in the classical theory of Hardy spaces and singular integrals can transplant to the function spaces associated with operators, such as [30–36].SupposeL is a nonnegative, self-adjoint differential operator, and L has H∞-calculus on L2(Rn). The kernel pt(x,y) of e-tL satisfies the Gaussian upper bound on Rn×Rn. Motivated by [17, 20, 37], etc, in this paper, we use the area integral function SL associated with the operator L to define the Herz-type Hardy space HK˙q,Lα,p(Rn). In order to obtain the atomic and molecular decompositions of the Herz-type Hardy space, the (α,q,M,L)-atom and the (α,q,M,L,ϵ)-molecule are introduced. By the method of the atomic and molecular decompositions of classical Herz-type Hardy spaces and the Hardy space HL1(Rn) associated with operators, we characterize HK˙q,Lα,p(Rn) spaces for atoms and molecules; that is, we prove the atomic and molecular decompositions of Herz-type Hardy spaces associated with the operator L. Finally, as applications, we prove some singular integral operators are bounded from HK˙q,Lα,p(Rn) to Herz spaces K˙qα,p(Rn) and also bounded on HK˙q,Lα,p(Rn).Throughout the paper, we always use the letterC to denote a positive constant, which may change from one to another and only depends on main parameters. We also use χE to denote the characteristic function of E which is the subset of Rn.


## 2. Preliminaries

For convenience, we recall the definitions of Herz and Herz-type Hardy spaces onRn. For details, we refer to [37, 38], etc.Letk∈Z, Bk={x∈Rn:x≤2k}, Ck=Bk\Bk-1 and χk=χCk.Definition 1 (see [38]).
Let0<α<∞, 0<p≤∞, 0<q≤∞.
( 1 ) The homogeneous Herz space K˙qα,p(Rn) is defined by (1)K˙qα,pRn=f∈LlocqRn\0:fK˙qα,pRn<∞, where (2)fK˙qα,pRn=∑k=-∞∞2kαpfχkLqRnp1/p.
( 2 ) The nonhomogeneous Herz space Kqα,p(Rn) is defined by (3)Kqα,pRn=f∈LlocqRn:fKqα,pRn<∞, where(4)fKqα,pRn=fχB0Lq+∑k=1∞2kαpfχkLqRnp1/p.
LetGf be the grand maximal function of f defined as Gf(x)=supφ∈AN⁡φ∇∗fx, where AN={φ∈S(Rn):supα,β≤N⁡xαDβφx≤1}, N>n+1.Definition 2 (see [38]).
Let0<α<∞,0<p<∞,0<q<∞.
( 1 ) The homogeneous Herz-Hardy space HK˙qα,p(Rn) is defined by (5)HK˙qα,pRn=f∈S′Rn:Gf∈K˙qα,pRn, Moreover,(6)fHK˙qα,pRn=GfK˙qα,pRn.(2) The nonhomogeneous Herz-Hardy space HKqα,p(Rn) is defined by(7)HKqα,pRn=f∈S′Rn:Gf∈Kqα,pRn. Moreover,(8)fHKqα,pRn=GfKqα,pRn.
Now, we introduce the Herz-type Hardy spaces associated with operators.
Suppose that the differential operatorL satisfies the following two assumptions.Assumption (A1). L is a nonnegative self-adjoint operator on L2(Rn), and has a bounded H∞-functional calculus in L2(Rn).Assumption (A2). Each of the heat semigroup e-tL generated by L has the kernel pt(x,y) which satisfies the following Gaussian upper bounds; i.e., there exist constants C,c>0 such that (9)ptx,y≤Ctn/2exp⁡-x-y2ct.Obviously, the typical second-order elliptic or subelliptic differential operators are to satisfy these assumptions (see for instance, [39]).Now we introduce the following lemmas which will be used in this paper.Lemma 3 (see [40, 41]).
LetL be a nonnegative self-adjoint operator satisfying Assumptions (A1) and (A2). For every j=0,1,2,…, there exist two positive constants Cj,cj such that the kernel pt,j(x,y) of the operator (t2L)je-t2L satisfies(10)pt,jx,y≤Cj4πtnexp⁡-x-y2cjt2,for all t>0 and almost every x,y∈Rn.Lemma 4 (see [19]).
Letφ∈C0∞(R) be even and suppφ⊆[-c0-1,c0-1]. Suppose Φ denotes the Fourier transform of φ. Then for each j=0,1,2,…, kernel K(t2L)jΦ(tL)(x,y) of (t2L)jΦ(tL) satisfies(11)supp⁡Kt2LjΦtL⊆x,y∈Rn×Rn:x-y≤tand(12)Kt2LjΦtLx,y≤Ct-n,for all t>0 and x,y∈Rn.Lemma 5 (see [19]).
Fors>0, we define (13)Fs≔ψ:C→Cmeasurable:ψz≤Czs1+z2s. Then for any nonzero function ψ∈F(s), κ={∫0∞|ψ(t)|2dt/t}1/2<∞.Lemma 6 (see [19]).
Letψ∈F(s). Then, for any f∈L2(Rn),(14)∫0∞ψtLfL2Rn2dtt1/2=κfL2Rn.Definition 7 (see [8]).
For anyf∈L1(Rn), the area integral function SL(f) associated with operators L is defined by(15)SLfx=∬x-y<tQt2fy2dydttn+11/2,where Qt2f(x)=t2Le-t2Lf(x), and L satisfies Assumptions (A1) and (A2).Thus, for any integerm>0, qtm, the kernel of Qtm satisfies |qtm|≤Ct-ns(|x-y|/t), where s is a positive decreasing function, satisfying limr→∞⁡rn+εsr=0, for any ε>0. Therefore, for convenience, in the following, we always set m=2 in (15).By the definition (15) and Assumptions (A1) and (A2), it is easy to check that for any f∈Lp(Rn), 1<p<∞; there exist C1,C2>0 such that (or see, for example, [11]):(16)C1fp≤SLfp≤C2fp.Definition 8.
Suppose0<α<∞,0<p<∞,1<q<∞. Let L satisfy Assumptions (A1) and (A2).
( 1 ) The homogeneous Herz-type Hardy space HK˙q,Lα,p(Rn) associated with the operator L is defined by(17)HK˙q,Lα,pRn=f∈S′Rn:SLf∈K˙qα,pRn. The norm of f in HK˙q,Lα,p(Rn) is(18)fHK˙q,Lα,pRn=SLfK˙qα,pRn.(2) The nonhomogeneous Herz-type Hardy space HKq,Lα,p(Rn) associated with the operator L is defined by(19)HKq,Lα,pRn=f∈S′Rn:SLf∈Kqα,pRn. The norm of f in HKq,Lα,p(Rn) is(20)fHKq,Lα,pRn=SLfKqα,pRn.
SinceK˙p0,p(Rn)=Kp0,p(Rn)=Lp(Rn), the Herz-type Hardy spaces associated with the operator introduced in Definition 8 are really the expansion of Hardy space associated with operators in [8].


## 3. The Decompositions of Herz-Type Hardy Spaces

In this section, we give the atomic decomposition and molecular decomposition of Herz-type Hardy spaces associated with the operatorL, respectively, which are the main results in this paper.


### 3.1. Atomic Decomposition of the Herz-Type Hardy Space

For the purpose of the atomic decomposition of the Herz-type Hardy space associated with operator, we first introduce the(α,q,M,L)-atom.Definition 9.
Let1<q<∞, 0<α<∞, M≥1. Set D(L)={u∈L2(Rn):Lu∈L2(Rn)}, where L satisfies Assumptions (A1) and (A2).
( 1 ) A function a(x)∈L2(Rn) is said to be an (α,q,M,L)-atom, if there exists b∈D(LM), such that(i)
a=LMb;(ii)
suppLjb⊂B(0,r),j=0,1,⋯,M;(iii)
(r2L)jbLq(Rn)≤r2M|B|-α/n,j=0,1,⋯,M;
B = B ( 0 , r ) = { x ∈ R n : | x | ≤ r }, r>0.
( 2 ) Function a(x)∈L2(Rn) is said to be a restrictive (α,q,M,L)-atom, if there exists b∈D(LM), satisfying (i), (ii), (iii), and B(0,r)={x∈Rn:|x|≤r},r≥1.The main result of this subsection is the following atomic decomposition of the Herz-type Hardy spaces associated with the operatorL. The part of the idea is from [1].Theorem 10.
Let0<p<∞, 1<q<∞, 0<α<n(1-1/q)+1. Suppose L satisfies Assumptions (A1) and (A2). Then, f∈HK˙q,Lα,p(Rn) if and only if there exist a family of (α,q,M,L)-atoms {ak} and a sequence of numbers {λk} such that f can be represented in the following form:(21)fx=∑k=-∞∞λkakx, and the sum converges in the sense of L2-norm, (∑k=-∞∞|λk|p)1/p<∞. Moreover,(22)fHK˙q,Lα,pRn~inf⁡∑k=-∞∞λkp1/p, where the infimum is taken over all of the decompositions of f.Proof.
First, we prove the theorem forq=2.
Necessity. Let φ and Φ be the same as those in Lemma 4. Set Ψ(x)=x2MΦ(x). Then by L2-functional calculus (see for example, [42]), for every f∈HK˙2,Lα,p(Rn), there is(23)fx=CΨ∫0∞ΨtLt2Le-t2Lfxdtt.
SetΩk={x∈Rn:SL(f)(x)>2k}, k∈Z. D denotes the collection of all dyadic cubes in Rn. Let Dk={Q∈D:|Q∩Ωk|>|Q|/2,|Q∩Ωk+1|≤|Q|/2}. Then, for any Q∈D, there exists only one k∈Z such that Q∈Dk. Let Dkl={Qkl∈Dk:Q∈Dk,Q∩Qkl≠∅,Q⊂Qkl}; i.e., Dkl denote the collection of maximal dyadic cubes in Dk. Set (24)Q^=y,t:y∈Q,lQ2<t<lQ, where l(Q) is the side length of Q.
Then, by (23), we have that (25)fx=∑k∑lCΨ∬Q^klΨtLx,yt2Le-t2Lfydydtt≔∑k,lλklaklx, where akl=LMbkl and(26)bklx=CΨλkl∬Q^klt2MΦtLx,yt2Le-t2Lfydydtt,λkl=Qklα/n∬Q^klt2Le-t2Lfy2dydtt1/2.
We will prove that, up to a normalization by a multiplicative constant, everyakl(x) is an (α,q,M,L)-atom.
Obviously, by Lemma4, we conclude that supp⁡(Ljbkl)⊂3Qkl, j=0,1,⋯,M.
Forh(x)∈L2(Rn), and hL2(Rn)≤1, then, by Hölder inequality together with Lemma 6, we have that(27)lQkl2LjbklL2Rn=sup∥h∥L2Rn≤1⁡∫RnlQkl2Ljbklxhxdx=sup∥h∥L2Rn≤1⁡CΨλkl∭Q^klt2MlQkl2LjΦtLx,yt2Le-t2Lfydydtthxdx=sup∥h∥L2Rn≤1⁡CΨλkl∬Q^klt2MlQkl2LjΦtLhyt2Le-t2Lfydydtt≤Csup∥h∥L2Rn≤1⁡1λkllQkl2M∬Q^klt2Le-t2Lfy2dydtt1/2∫Rn+1t2LjΦtL2hydydtt1/2=Csup∥h∥L2Rn≤1⁡lQkl2Mλkl∬Q^klt2Le-t2Lfy2dydtt1/2hL2Rn=ClQkl2MQkl-α/n.
Furthermore, we prove the following estimate:(28)∑k,lλklp1/p≤CfHK˙2,Lα,pRn.Noting the definition of λkl in (26) and Definition 8 for HK˙2,Lα,p(Rn), it means that we should establish the following inequality:(29)∑k,lQklαp/n∬Q^klt2Le-t2Lfy2dydttp/2≤C∑k,lQklαp/n∫RnSLfχl,k2dxp/2,where χl,k=χEkl is the characteristic function of Ekl=Qkl\Qk-1l.
It is sufficient to show that(30)∬Q^klt2Le-t2Lfy2dydtt≤C∫EklSLfx2dx.In fact, if (y,t)∈Q^kl, then y∈Qkl, l(Qkl)/2<t<l(Qkl). Let χ(x,y,t) denote the characteristic function of {(x,y,t):x∈Ekl,|x-y|<t}. Thus, by the definition of Dk, we obtain that (31)∫Rnχx,y,tdx≥Ctn. Therefore, (32)∬Q^klt2Le-t2Lfy2dydtt≤C∬Q^kl∫Eklχx,y,tt2Le-t2Lfy2dxdydttn+1≤C∫Ekl∫Rn+1χx,y,tt2Le-t2Lfy2dydttn+1dx≤C∫EklSLfx2dx. Hence, the necessity is proved.
Sufficiency. Let f(x)=∑k=-∞∞λkak(x), where every ak is an (α,q,M,L)-atom. We will prove the sufficiency for two situations: 0<p≤1 and 1<p<∞.
If0<p≤1, then, to prove f∈HK˙2,Lα,p(Rn), it is only need to show that, for any (α,q,M,L)-atom a, there exits a constant C>0 independent of a such that (33)SLaK˙2α,pRn≤C.In fact, then, we have (34)fHK˙2,Lα,pRnp=SLfK˙2α,pRnp≤∑k=-∞∞λkpSLakK˙2α,pRnp≤C∑k=-∞∞λkp.
Suppose thata is an (α,q,M,L)-atom with a=LMb and for any m (m=0,1,2,…,M) suppLmb⊂Bk0=B(0,2k0), k0∈Z+. Then (35)SLaK˙2α,pRnp=∑k=-∞∞Bkαp/nSLaχkL2Rnp=∑k=-∞k0+1Bkαp/nSLaχkL2Rnp+∑k=k0+2∞Bkαp/nSLaχkL2Rnp≔I1+I2. For I1, L2 boundedness of SL and the size condition of atom tell us (36)I1≤C∑k=-∞k0+1Bkαp/nSLaχkL2Rnp≤C∑k=-∞k0+1Bkαp/naL2Rnp≤C∑k=-∞k0+1Bkαp/nBk0-αp/n≤C∑k=-∞k0+12k-k0αp≤C. In order to estimate I2, we write (37)SLa2x=∫0∞∫x-y<tt2Le-t2Lay2dydttn+1=∫02k0+∫2k0∞∫x-y<tt2Le-t2Lay2dydttn+1≔I2,1+I2,2. For I2,1, noting that k>k0+1 and x-y<t<2k0, if x∈Ck, z∈Bk0, then |y-z|≥|x-z|-|x-y|>2k-1-2k0≥C2k. Thus, by Lemma 3, we can have that(38)I2,1≤C∫02k0∫x-y<t∫Rntt+y-zn+1azdz2dydttn+1≤CaL1Rn22k2n+1∫02k0∫x-y<tt2dydttn+1≤CaL2Rn2Bk02k2n+122k0=CBk0-2α/nBk0Bk-22-2k22k0.For I2,2, noting that |x-y|<t, if x∈Ck, z∈Bk0, then |x-z|≤|x-y|+|y-z|<t+|y-z|. So that t+|y-z|>C2k holds true. Therefore, we can obtain that(39)I2,2=∫2k0∞∫x-y<tt2Le-t2LLMb2dydttn+1=∫2k0∞∫x-y<tt2LM+1e-t2Lby2dydttn+4M+1≤C∫2k0∞∫x-y<t∫Rntt+y-zn+1bzdz2dydttn+4M+1≤CbL1Rn22k2n+1∫2k0∞dtt4M-1=CBk0-2α/nBk0Bk-22-k22k0.Hence, combining (38) and (39), one can have (40)I2=∑k=k0+2∞Bkαp/nSLaχkL2Rnp≤C∑k=k0+2∞Bkαp/nBk0-αp/nBk-p/2Bk0p/22k0p2kp≤C∑k=k0+2∞2k-k0α-n/2-1p≤C.If 1<p<∞, then (41)SLfK˙2α,pRnp≤∑k=-∞∞Bkαp/n∑l=-∞∞λlSLalχkL2Rnp=C∑k=-∞∞Bkαp/n∑l=k-1∞λlSLalχkL2Rnp+C∑k=-∞∞Bkαp/n∑l=-∞k-2λlSLalχkL2Rnp≔II1+II2. For II1, L2 boundedness of SL and the Hölder inequality tell us (42)II1≤C∑k=-∞∞Bkαp/n∑l=k-1∞λlalL2Rnp≤C∑k=-∞∞Bkαp/n∑l=k-1∞λlBl-α/np≤C∑k=-∞∞Bkαp/n∑l=k-1∞λlpBl-αp/2n∑l=k-1∞Bl-αp′/2np/p′≤C∑k=-∞∞Bkαp/2n∑l=k-1∞λlpBl-αp/2n=C∑l=-∞∞λlpBl-αp/2n∑k=-∞l+1Bkαp/2n=C∑l=-∞∞λlp. For II2, similar to the estimate of SL(a)(x), we can obtain the estimates of SL(al) as (38) and (39). Thus, using the Hölder inequality, we have that (43)II2≤∑k=-∞∞Bkαp/n∑l=-∞k-2λlBl-α/nBl1/2Bk-1/22l2-kp≤C∑k=-∞∞∑l=-∞k-2λl2k-lα-n/2-1p≤C∑k=-∞∞∑l=-∞k-2λlp2k-lα-n/2-1p/2∑l=-∞k-22k-lα-n/2-1p′/2p/p′≤C∑l=-∞∞λlp∑k=l+2+∞2k-lα-n/2-1p/2=C∑l=-∞∞λlp. The sufficiency is proved. Then the proof of Theorem 10 for q=2 is finished.
Ifq≠2, the proof that is exactly similar to the situation of q=2, we only need to slightly modify some formulas above. We should set(44)λkl=Qklα/n+1/q-1/2∬Q^klt2Le-t2Lfy2dydtt1/2. Inequalities (29) and (30) are replaced with(45)∑k,lQklαp/nQkl1/q-1/2p∬Q^klt2Le-t2Lfy2dydttp/2≤C∑k,lQklαp/nSLfx·χl,kLqRnpand(46)Qkl1/q-1/2∬Q^klt2Le-t2Lfy2dydtt1/2≤CSLf·χl,kLqRn,respectively. To obtain inequality (46), there is (47)SLf·χl,kLqRn=∫Ekl∬x-y<tt2Le-t2Lfy2dydttn+1q/2dx1/q≥C∫Ekl∬x-y<tt2Le-t2Lfy2dydttq/21Qklq/2dx1/q≥CQkl-1/2∫Ekl∬x-y<tt2Le-t2Lfy2dydttdxq/21/q≥CQkl1/q-1/2∬Q^klt2Le-t2Lfy2dydtt1/2.Other details are omitted.For the nonhomogeneous Herz-type Hardy spaceHKq,Lα,p(Rn) associated with the operator L, there is the same result as follows.Theorem 11.
Let0<p<∞, 1<q<∞, 0<α<n(1-1/q)+1. Suppose L satisfies Assumptions (A1) and (A2). Then, f∈HKq,Lα,p(Rn) if and only if there exist a family of the restrictive (α,q,M,L)-atoms {ak}k=0+∞ and a sequence of numbers {λk}k=0+∞ such that f can be represented in the following form:(48)fx=∑k=0+∞λkakx, and the sum converges in the sense of L2-norm, (∑k=0+∞|λk|p)1/p<∞. Moreover,(49)fHKq,Lα,pRn~inf⁡∑k=0+∞λkp1/p, where the infimum is taken over all of the decompositions of f.


### 3.2. Molecular Decomposition of the Herz-Type Hardy Space

In this subsection, we first introduce(α,q,M,L,ϵ)-molecule in the following; then we give the molecular decomposition of the Herz-type Hardy space associated with operator.Definition 12.
Let1<q<∞, 0<α<∞, M≥1. Set D(L)={u∈L2(Rn):Lu∈L2(Rn)}, where L satisfies Assumptions (A1) and (A2).
( 1 ) A function a(x)∈L2(Rn) is said to be an (α,q,M,L,ϵ)-molecule, if there exists b∈D(LM), such that(i)
a=LMb;(ii)
(r2L)jbLq(Sk(B))≤2-kϵr2M|2kB|-α/n,j=0,1,⋯,M and k=0,1,⋯,
whereB=B(0,r)={x∈Rn:|x|≤r}, r>0; and(50)S0B=B,SkB=2kB\2k-1Bfork∈N.(2) Function a(x)∈L2(Rn) is said to be a restrictive (α,q,M,L,ϵ)-molecule, if there exists b∈D(LM), satisfying (i), (ii), and B(0,r)={x∈Rn:|x|≤r},r≥1.It is not difficult to check that an(α,q,M,L)-atom associated with ball B is also an (α,q,M,L,ϵ)-molecule associated with the same ball B.The following molecular decomposition of the Herz-type Hardy spaces associated with the operatorL is the main result in this subsection.Theorem 13.
Let0<p<∞, 1<q<∞, 0<α<n(1-1/q)+1. Suppose L satisfies Assumptions (A1) and (A2). Then, f∈HK˙q,Lα,p(Rn) if and only if there exist a family of (α,q,M,L,ϵ)-molecules {ak} and a sequence of numbers {λk} such that f can be represented in the following form:(51)fx=∑k=-∞∞λkakx, and the sum converges in the sense of L2-norm, (∑k=-∞∞|λk|p)1/p<∞. Moreover,(52)fHK˙q,Lα,pRn~inf⁡∑k=-∞∞λkp1/p, where the infimum is taken over all of the decompositions of f.Proof.
(i) The proof of necessity is a direct consequence of the necessity in Theorem10, since an (α,q,M,L)-atom is also an (α,q,M,L,ϵ)-molecule for all ϵ>0.
(ii) The proof of sufficiency is similar to that of the sufficiency in Theorem10. The main difference is that the support of (α,q,M,L,ϵ)-molecule is not the ball B. However, we can overcome this difficulty by decomposing Rn into annuli associated with the ball B, then using the same argument as in Theorem 10 to get sufficiency. We omit the details here.Similarly, for the nonhomogeneous Herz-type Hardy space associated with operatorHKq,Lα,p(Rn), there is the same result as follows.Theorem 14.
Let0<p<∞, 1<q<∞, 0<α<n(1-1/q)+1. Suppose L satisfies Assumptions (A1) and (A2). Then, f∈HKq,Lα,p(Rn) if and only if there exist a family of the restrictive (α,q,M,L,ϵ)-molecules {ak}k=0+∞ and a sequence of numbers {λk}k=0+∞ such that f can be represented in the following form:(53)fx=∑k=0+∞λkakx, and the sum converges in the sense of L2-norm, (∑k=0+∞|λk|p)1/p<∞. Moreover,(54)fHKq,Lα,pRn~inf⁡∑k=0+∞λkp1/p, where the infimum is taken over all of the decompositions of f.


## 3.1. Atomic Decomposition of the Herz-Type Hardy Space

For the purpose of the atomic decomposition of the Herz-type Hardy space associated with operator, we first introduce the(α,q,M,L)-atom.Definition 9.
Let1<q<∞, 0<α<∞, M≥1. Set D(L)={u∈L2(Rn):Lu∈L2(Rn)}, where L satisfies Assumptions (A1) and (A2).
( 1 ) A function a(x)∈L2(Rn) is said to be an (α,q,M,L)-atom, if there exists b∈D(LM), such that(i)
a=LMb;(ii)
suppLjb⊂B(0,r),j=0,1,⋯,M;(iii)
(r2L)jbLq(Rn)≤r2M|B|-α/n,j=0,1,⋯,M;
B = B ( 0 , r ) = { x ∈ R n : | x | ≤ r }, r>0.
( 2 ) Function a(x)∈L2(Rn) is said to be a restrictive (α,q,M,L)-atom, if there exists b∈D(LM), satisfying (i), (ii), (iii), and B(0,r)={x∈Rn:|x|≤r},r≥1.The main result of this subsection is the following atomic decomposition of the Herz-type Hardy spaces associated with the operatorL. The part of the idea is from [1].Theorem 10.
Let0<p<∞, 1<q<∞, 0<α<n(1-1/q)+1. Suppose L satisfies Assumptions (A1) and (A2). Then, f∈HK˙q,Lα,p(Rn) if and only if there exist a family of (α,q,M,L)-atoms {ak} and a sequence of numbers {λk} such that f can be represented in the following form:(21)fx=∑k=-∞∞λkakx, and the sum converges in the sense of L2-norm, (∑k=-∞∞|λk|p)1/p<∞. Moreover,(22)fHK˙q,Lα,pRn~inf⁡∑k=-∞∞λkp1/p, where the infimum is taken over all of the decompositions of f.Proof.
First, we prove the theorem forq=2.
Necessity. Let φ and Φ be the same as those in Lemma 4. Set Ψ(x)=x2MΦ(x). Then by L2-functional calculus (see for example, [42]), for every f∈HK˙2,Lα,p(Rn), there is(23)fx=CΨ∫0∞ΨtLt2Le-t2Lfxdtt.
SetΩk={x∈Rn:SL(f)(x)>2k}, k∈Z. D denotes the collection of all dyadic cubes in Rn. Let Dk={Q∈D:|Q∩Ωk|>|Q|/2,|Q∩Ωk+1|≤|Q|/2}. Then, for any Q∈D, there exists only one k∈Z such that Q∈Dk. Let Dkl={Qkl∈Dk:Q∈Dk,Q∩Qkl≠∅,Q⊂Qkl}; i.e., Dkl denote the collection of maximal dyadic cubes in Dk. Set (24)Q^=y,t:y∈Q,lQ2<t<lQ, where l(Q) is the side length of Q.
Then, by (23), we have that (25)fx=∑k∑lCΨ∬Q^klΨtLx,yt2Le-t2Lfydydtt≔∑k,lλklaklx, where akl=LMbkl and(26)bklx=CΨλkl∬Q^klt2MΦtLx,yt2Le-t2Lfydydtt,λkl=Qklα/n∬Q^klt2Le-t2Lfy2dydtt1/2.
We will prove that, up to a normalization by a multiplicative constant, everyakl(x) is an (α,q,M,L)-atom.
Obviously, by Lemma4, we conclude that supp⁡(Ljbkl)⊂3Qkl, j=0,1,⋯,M.
Forh(x)∈L2(Rn), and hL2(Rn)≤1, then, by Hölder inequality together with Lemma 6, we have that(27)lQkl2LjbklL2Rn=sup∥h∥L2Rn≤1⁡∫RnlQkl2Ljbklxhxdx=sup∥h∥L2Rn≤1⁡CΨλkl∭Q^klt2MlQkl2LjΦtLx,yt2Le-t2Lfydydtthxdx=sup∥h∥L2Rn≤1⁡CΨλkl∬Q^klt2MlQkl2LjΦtLhyt2Le-t2Lfydydtt≤Csup∥h∥L2Rn≤1⁡1λkllQkl2M∬Q^klt2Le-t2Lfy2dydtt1/2∫Rn+1t2LjΦtL2hydydtt1/2=Csup∥h∥L2Rn≤1⁡lQkl2Mλkl∬Q^klt2Le-t2Lfy2dydtt1/2hL2Rn=ClQkl2MQkl-α/n.
Furthermore, we prove the following estimate:(28)∑k,lλklp1/p≤CfHK˙2,Lα,pRn.Noting the definition of λkl in (26) and Definition 8 for HK˙2,Lα,p(Rn), it means that we should establish the following inequality:(29)∑k,lQklαp/n∬Q^klt2Le-t2Lfy2dydttp/2≤C∑k,lQklαp/n∫RnSLfχl,k2dxp/2,where χl,k=χEkl is the characteristic function of Ekl=Qkl\Qk-1l.
It is sufficient to show that(30)∬Q^klt2Le-t2Lfy2dydtt≤C∫EklSLfx2dx.In fact, if (y,t)∈Q^kl, then y∈Qkl, l(Qkl)/2<t<l(Qkl). Let χ(x,y,t) denote the characteristic function of {(x,y,t):x∈Ekl,|x-y|<t}. Thus, by the definition of Dk, we obtain that (31)∫Rnχx,y,tdx≥Ctn. Therefore, (32)∬Q^klt2Le-t2Lfy2dydtt≤C∬Q^kl∫Eklχx,y,tt2Le-t2Lfy2dxdydttn+1≤C∫Ekl∫Rn+1χx,y,tt2Le-t2Lfy2dydttn+1dx≤C∫EklSLfx2dx. Hence, the necessity is proved.
Sufficiency. Let f(x)=∑k=-∞∞λkak(x), where every ak is an (α,q,M,L)-atom. We will prove the sufficiency for two situations: 0<p≤1 and 1<p<∞.
If0<p≤1, then, to prove f∈HK˙2,Lα,p(Rn), it is only need to show that, for any (α,q,M,L)-atom a, there exits a constant C>0 independent of a such that (33)SLaK˙2α,pRn≤C.In fact, then, we have (34)fHK˙2,Lα,pRnp=SLfK˙2α,pRnp≤∑k=-∞∞λkpSLakK˙2α,pRnp≤C∑k=-∞∞λkp.
Suppose thata is an (α,q,M,L)-atom with a=LMb and for any m (m=0,1,2,…,M) suppLmb⊂Bk0=B(0,2k0), k0∈Z+. Then (35)SLaK˙2α,pRnp=∑k=-∞∞Bkαp/nSLaχkL2Rnp=∑k=-∞k0+1Bkαp/nSLaχkL2Rnp+∑k=k0+2∞Bkαp/nSLaχkL2Rnp≔I1+I2. For I1, L2 boundedness of SL and the size condition of atom tell us (36)I1≤C∑k=-∞k0+1Bkαp/nSLaχkL2Rnp≤C∑k=-∞k0+1Bkαp/naL2Rnp≤C∑k=-∞k0+1Bkαp/nBk0-αp/n≤C∑k=-∞k0+12k-k0αp≤C. In order to estimate I2, we write (37)SLa2x=∫0∞∫x-y<tt2Le-t2Lay2dydttn+1=∫02k0+∫2k0∞∫x-y<tt2Le-t2Lay2dydttn+1≔I2,1+I2,2. For I2,1, noting that k>k0+1 and x-y<t<2k0, if x∈Ck, z∈Bk0, then |y-z|≥|x-z|-|x-y|>2k-1-2k0≥C2k. Thus, by Lemma 3, we can have that(38)I2,1≤C∫02k0∫x-y<t∫Rntt+y-zn+1azdz2dydttn+1≤CaL1Rn22k2n+1∫02k0∫x-y<tt2dydttn+1≤CaL2Rn2Bk02k2n+122k0=CBk0-2α/nBk0Bk-22-2k22k0.For I2,2, noting that |x-y|<t, if x∈Ck, z∈Bk0, then |x-z|≤|x-y|+|y-z|<t+|y-z|. So that t+|y-z|>C2k holds true. Therefore, we can obtain that(39)I2,2=∫2k0∞∫x-y<tt2Le-t2LLMb2dydttn+1=∫2k0∞∫x-y<tt2LM+1e-t2Lby2dydttn+4M+1≤C∫2k0∞∫x-y<t∫Rntt+y-zn+1bzdz2dydttn+4M+1≤CbL1Rn22k2n+1∫2k0∞dtt4M-1=CBk0-2α/nBk0Bk-22-k22k0.Hence, combining (38) and (39), one can have (40)I2=∑k=k0+2∞Bkαp/nSLaχkL2Rnp≤C∑k=k0+2∞Bkαp/nBk0-αp/nBk-p/2Bk0p/22k0p2kp≤C∑k=k0+2∞2k-k0α-n/2-1p≤C.If 1<p<∞, then (41)SLfK˙2α,pRnp≤∑k=-∞∞Bkαp/n∑l=-∞∞λlSLalχkL2Rnp=C∑k=-∞∞Bkαp/n∑l=k-1∞λlSLalχkL2Rnp+C∑k=-∞∞Bkαp/n∑l=-∞k-2λlSLalχkL2Rnp≔II1+II2. For II1, L2 boundedness of SL and the Hölder inequality tell us (42)II1≤C∑k=-∞∞Bkαp/n∑l=k-1∞λlalL2Rnp≤C∑k=-∞∞Bkαp/n∑l=k-1∞λlBl-α/np≤C∑k=-∞∞Bkαp/n∑l=k-1∞λlpBl-αp/2n∑l=k-1∞Bl-αp′/2np/p′≤C∑k=-∞∞Bkαp/2n∑l=k-1∞λlpBl-αp/2n=C∑l=-∞∞λlpBl-αp/2n∑k=-∞l+1Bkαp/2n=C∑l=-∞∞λlp. For II2, similar to the estimate of SL(a)(x), we can obtain the estimates of SL(al) as (38) and (39). Thus, using the Hölder inequality, we have that (43)II2≤∑k=-∞∞Bkαp/n∑l=-∞k-2λlBl-α/nBl1/2Bk-1/22l2-kp≤C∑k=-∞∞∑l=-∞k-2λl2k-lα-n/2-1p≤C∑k=-∞∞∑l=-∞k-2λlp2k-lα-n/2-1p/2∑l=-∞k-22k-lα-n/2-1p′/2p/p′≤C∑l=-∞∞λlp∑k=l+2+∞2k-lα-n/2-1p/2=C∑l=-∞∞λlp. The sufficiency is proved. Then the proof of Theorem 10 for q=2 is finished.
Ifq≠2, the proof that is exactly similar to the situation of q=2, we only need to slightly modify some formulas above. We should set(44)λkl=Qklα/n+1/q-1/2∬Q^klt2Le-t2Lfy2dydtt1/2. Inequalities (29) and (30) are replaced with(45)∑k,lQklαp/nQkl1/q-1/2p∬Q^klt2Le-t2Lfy2dydttp/2≤C∑k,lQklαp/nSLfx·χl,kLqRnpand(46)Qkl1/q-1/2∬Q^klt2Le-t2Lfy2dydtt1/2≤CSLf·χl,kLqRn,respectively. To obtain inequality (46), there is (47)SLf·χl,kLqRn=∫Ekl∬x-y<tt2Le-t2Lfy2dydttn+1q/2dx1/q≥C∫Ekl∬x-y<tt2Le-t2Lfy2dydttq/21Qklq/2dx1/q≥CQkl-1/2∫Ekl∬x-y<tt2Le-t2Lfy2dydttdxq/21/q≥CQkl1/q-1/2∬Q^klt2Le-t2Lfy2dydtt1/2.Other details are omitted.For the nonhomogeneous Herz-type Hardy spaceHKq,Lα,p(Rn) associated with the operator L, there is the same result as follows.Theorem 11.
Let0<p<∞, 1<q<∞, 0<α<n(1-1/q)+1. Suppose L satisfies Assumptions (A1) and (A2). Then, f∈HKq,Lα,p(Rn) if and only if there exist a family of the restrictive (α,q,M,L)-atoms {ak}k=0+∞ and a sequence of numbers {λk}k=0+∞ such that f can be represented in the following form:(48)fx=∑k=0+∞λkakx, and the sum converges in the sense of L2-norm, (∑k=0+∞|λk|p)1/p<∞. Moreover,(49)fHKq,Lα,pRn~inf⁡∑k=0+∞λkp1/p, where the infimum is taken over all of the decompositions of f.


## 3.2. Molecular Decomposition of the Herz-Type Hardy Space

In this subsection, we first introduce(α,q,M,L,ϵ)-molecule in the following; then we give the molecular decomposition of the Herz-type Hardy space associated with operator.Definition 12.
Let1<q<∞, 0<α<∞, M≥1. Set D(L)={u∈L2(Rn):Lu∈L2(Rn)}, where L satisfies Assumptions (A1) and (A2).
( 1 ) A function a(x)∈L2(Rn) is said to be an (α,q,M,L,ϵ)-molecule, if there exists b∈D(LM), such that(i)
a=LMb;(ii)
(r2L)jbLq(Sk(B))≤2-kϵr2M|2kB|-α/n,j=0,1,⋯,M and k=0,1,⋯,
whereB=B(0,r)={x∈Rn:|x|≤r}, r>0; and(50)S0B=B,SkB=2kB\2k-1Bfork∈N.(2) Function a(x)∈L2(Rn) is said to be a restrictive (α,q,M,L,ϵ)-molecule, if there exists b∈D(LM), satisfying (i), (ii), and B(0,r)={x∈Rn:|x|≤r},r≥1.It is not difficult to check that an(α,q,M,L)-atom associated with ball B is also an (α,q,M,L,ϵ)-molecule associated with the same ball B.The following molecular decomposition of the Herz-type Hardy spaces associated with the operatorL is the main result in this subsection.Theorem 13.
Let0<p<∞, 1<q<∞, 0<α<n(1-1/q)+1. Suppose L satisfies Assumptions (A1) and (A2). Then, f∈HK˙q,Lα,p(Rn) if and only if there exist a family of (α,q,M,L,ϵ)-molecules {ak} and a sequence of numbers {λk} such that f can be represented in the following form:(51)fx=∑k=-∞∞λkakx, and the sum converges in the sense of L2-norm, (∑k=-∞∞|λk|p)1/p<∞. Moreover,(52)fHK˙q,Lα,pRn~inf⁡∑k=-∞∞λkp1/p, where the infimum is taken over all of the decompositions of f.Proof.
(i) The proof of necessity is a direct consequence of the necessity in Theorem10, since an (α,q,M,L)-atom is also an (α,q,M,L,ϵ)-molecule for all ϵ>0.
(ii) The proof of sufficiency is similar to that of the sufficiency in Theorem10. The main difference is that the support of (α,q,M,L,ϵ)-molecule is not the ball B. However, we can overcome this difficulty by decomposing Rn into annuli associated with the ball B, then using the same argument as in Theorem 10 to get sufficiency. We omit the details here.Similarly, for the nonhomogeneous Herz-type Hardy space associated with operatorHKq,Lα,p(Rn), there is the same result as follows.Theorem 14.
Let0<p<∞, 1<q<∞, 0<α<n(1-1/q)+1. Suppose L satisfies Assumptions (A1) and (A2). Then, f∈HKq,Lα,p(Rn) if and only if there exist a family of the restrictive (α,q,M,L,ϵ)-molecules {ak}k=0+∞ and a sequence of numbers {λk}k=0+∞ such that f can be represented in the following form:(53)fx=∑k=0+∞λkakx, and the sum converges in the sense of L2-norm, (∑k=0+∞|λk|p)1/p<∞. Moreover,(54)fHKq,Lα,pRn~inf⁡∑k=0+∞λkp1/p, where the infimum is taken over all of the decompositions of f.


## 4. Boundedness of Singular Integral Operators

This section is based on the decompositions off∈HK˙q,Lα,p(Rn) in the previous section; as applications, we give some boundedness of sublinear operators satisfying certain conditions on Herz-type Hardy spaces associated with operator.Theorem 15.
Let0<p<∞, 1<q<∞, 0<α<n(1-1/q). If a sublinear operator T satisfies that(i)
T is bounded on Lq(Rn);(ii)
Tg satisfies the size condition(55)Tgx≤Cx-ng1,for suitable function g with dist(x,suppg)≥x/2.
ThenT is bounded from HK˙q,Lα,pRn to K˙qα,p(Rn), that is, (56)TfK˙qα,pRn≤CfHK˙q,Lα,pRn.Proof.
Supposef∈HK˙q,Lα,p(Rn). By Theorem 10, we have(57)fx=∑j=-∞∞λjajx, where each aj is a (α,q,M,L)-atom with aj=LMbj, and suppLmbj⊂Bj, m=0,1,…,M, fHK˙q,Lα,p(Rn)~inf⁡(∑j=-∞∞‍|λj|p)1/p.
Thus,(58)TfK˙qα,pRnp=∑k=-∞∞Bkαp/nTf·χkLqRnp≤C∑k=-∞∞Bkαp/n∑j=-∞k-2λj∥Taj·χk∥LqRnp+C∑k=-∞∞Bkαp/n∑j=k-1∞λjTaj·χkLqRnp≔J1+J2First, we estimate J2. By the boundedness of T in Lq(Rn), we can infer that (59)J2≤C∑k=-∞∞Bkαp/n∑j=k-1∞λjTaj·χkLqRnp≤C∑k=-∞∞∑j=k-1∞λj2k-jαp.Therefore, if 0<p≤1, then, by the Jensen inequality, we have (60)J2≤C∑k=-∞∞∑j=k-1∞λjp2k-jαp≤C∑j=-∞∞λjp∑k≤j+12k-jαp≤C∑j=-∞∞λjp. If 1<p<∞, let 1/p+1/p′=1. Then we can obtain (61)J2≤C∑k=-∞∞∑j=k-1∞λjp2k-jαp/2∑j=k-1∞2k-jαp′/2p/p′≤C∑j=-∞∞λjp∑k≤j+12k-jαp/2≤C∑j=-∞∞λjp. Hence, J2≤CfHK˙q,Lα,p(Rn).
Second, we estimateJ1. By the Hölder inequality and (55), we obtain (62)Taj·χkLqRn≤C∫Ckx-nq∫Bjajdyqdx1/q≤C2-nkBk1/q∫Bjajqdy∫Bjdyq/q′1/q≤C2-nkBk1/qBj1/q′ajLqRn≤C2j-kn/q′Bj-α/n. Thus,(63)J1≤C∑k=-∞∞∑j=-∞k-2λj2j-kn/q′-αp. Therefore, if 0<p≤1, by the Jensen inequality, we have (64)J1≤C∑k=-∞∞∑j=-∞k-2λjp2j-kn/q′-αp≤C∑j=-∞∞λjp∑k≥j2j-kn/q′-αp≤C∑j=-∞∞λjp. If 1<p<∞, by the Hölder inequality, we obtain(65)J1≤C∑k=-∞∞∑j=-∞k-2λjp2j-kn/q′-αp/2∑j=-∞k-22j-kn/q′-αp′/21/p′≤C∑k=-∞∞∑j=-∞k-2λjp2j-kn/q′-αp/2≤C∑j=-∞∞λjp∑k≥j2j-kn/q′-αp/2≤C∑j=-∞∞λjp. Hence,(66)TfK˙qα,pRn≤CfHK˙q,Lα,pRn.Theorem 16.
Let0<p<∞, 1<q<∞, 0<α<n(1-1/q). Suppose that T is a sublinear operator as Theorem 15 and T and L are commutative. Then T is bounded on HK˙q,Lα,p(Rn).Proof.
Supposea is an (α,q,M,L)-atom. According to Definition 9, there exists b∈D(LM), such that(i)
a=LMb;(ii)
supp⁡Ljb⊂B(0,2l),j=0,1,⋯,M;(iii)
r2LjbLq(Rn)≤r2MB-α/n,j=0,1,⋯,M.
T a being an (α,q,M,L,ϵ)-molecule only needs to be prove, such that(1)
Ta=LMTb;(2)
(r2L)jTbLq(Sk(B))≤2-kϵr2M2kB-α/n,j=0,1,⋯,M and k=0,1,⋯.
In fact, it is enough to check(2). For any j=0,1,⋯,M and k=0,1,⋯, we have(67)r2LjTbLqSkB=∫2kB\2k-1BTr2Ljbqdx1/q≤∫2kB\2k-1Bx-nq∫Br2Ljbdyqdx1/q≤C2-k+l-1n2kB1/qr2LjLqRnB1/q′≤C2-kn2-ln2nk+l/qB-α/nr2M2ln/q′≤C2-kn1-1/q-α/n2kB-α/nr2M. Thus, we complete the proof of Theorem 16.



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*Source: 1296837-2018-07-17.xml*