Liguang LIU Dachun YANG

1 Introduction

The local weight class was introduced by Rychkov[18].Recall that,forp∈(1,∞),the local weight classconsists of all non-negative locally integrable functionswsuch that

where the supremum is taken over all cubes of Rnwith then-dimensional Lebesgue measure no more than 1 and with sides parallelling to the coordinate axis.Ifp=1,then the classconsists of all non-negative locally integrable functionswsuch that

DefineFor anyq∈[1,∞],letqw:=which is called the critical index of the local weightw.Observe that the classconsists of non-doubling weights,which may grow or decrease exponentially at infinity.Apart from the well-known Muckenhoupt weight class,an important example ofw∈is from Triebel[24,Chapter 6],wherein the author investigated weighted inhomogeneous Besov and Triebel-Lizorkin spaces associated to a weightwsatisfying that there existβ∈(0,1]andC∈(0,∞)such that,for allx,y∈Rn,0

Rychkov[18]introduced and studied the inhomogeneous Besov and Triebel-Lizorkin spaces associated to a weightw∈Izuki and Sawano[14–16]then investigated the wavelet characterizations of these function spaces.Also,Tang[22]established the maximal function characterization of the weighted local Hardy spaceswithw∈which is an extension of the results of Bui[1]and Goldberg[7].Boundedness of some strongly singular integrals,pseudo-differential operators and their commutators on the weighted local Hardy spaceswere also studied in[22–23].For generalizations of the results in[22]to some Orlicz-type local Hardy spaces associated to the weightw∈(see[25]).It should be mentioned that there are many works concerning the(in)homogeneous Besov or Triebel-Lizorkin spaces associated to the classical Muckenhoupt weights;see,for instance,[9–13,24]and their references.

The main aim of this paper is to characterize the inhomogeneous Triebel-Lizorkin spaceswith local weightw∈in the sense of Rychkov[18](see also Definition 1.1 below)by using atoms completely analogous to the classical atoms of Hardy spaces.In other words,fors∈R,p∈(0,1],q∈[p,∞)andw∈we prove,in Theorem 1.2 below,that an elementf∈if and only if it can be written as a linear combination of these weighted atoms with the coefficients belonging to.To this end,we first establish the Lusin-area function characterization ofin Theorem 1.1 below.Moreover,finite atomic decompositions for smooth functions inare presented in Theorem 1.3 below.This allows us to deduce the following boundedness criteria in Theorem 1.4 below:If a(sub)linear operator maps atoms,which are infinitely differentiable,ofinto a(quasi-)Banach space uniformly,then it extends to a bounded(sub)linear operator on the wholeAs an application,the boundedness of the local Riesz operator on the spaceis obtained.It is expectable that Theorem 1.4 may have further more applications,say,in the study of the boundedness of operators on(see,for example,[2–3,17]).

It should be mentioned that the definition of atoms of(see Definition 1.2 below)used by us throughout this paper is inspired by Han,Paluszyski and Weiss[8],in which atomic characterizations for the classical non-weighted homogeneous Triebel-Lizorkin spacewere established,wheres∈R,p∈(0,1]andq∈[p,∞).We also remark that atomic decompositions of the Triebel-Lizorkin spaceswith local weights were also considered in[15],by using the machinery of theφ-transform of Frazier-Jawerth in[4–5].The advantage of the atoms used in this paper is that it is more convenient for applications in the study on the boundedness of operators.

To recall the inhomogeneous Triebel-Lizorkin spaces with local weights introduced in[18],we need the following notation.Letbe the set of all infinitely differentiable functions on Rnwith compact support.Endowwith the strict inductive topology,which is denoted byD(Rn)and whose dual space byAs in[18],letbe the space of allf∈(Rn)such that there exist positive constantsAfandNfsuch that for allφ∈D(Rn),|≤Afsup{:x∈Rn,|α|≤Nf}.Forp∈(0,∞],we denote bythe weighted Lebesgue space which consists of all functionsfsuch thatf:=|f(x)|pw(x)dx<∞,and bythe weighted weak-type Lebesgue space which consists of all functionsfsuch thatf:=({x∈Rn:|f(x)|>t})<∞.For anys∈R,we denote bysthe maximal integer no more thans.

Definition 1.1Let ws∈R,p∈(0,∞)and q∈(0,∞].Suppose that φ0∈D(Rn)and φ:=φ0−satisfies that

For j∈N,set φj:=The inhomogeneous Triebel-Lizorkin spaceis defined to be the collection of all f∈such that

with a usual modification made when q=∞.

Notice thatwithw∈s∈R,p∈(0,∞)andq∈(0,∞],are complete(quasi-)Banach spaces(see[18,Lemma 2.15]).By[18,Theorem 2.5],there exist positive constantsA0andB0,depending only ons,p,qandw,such that,whenA≥A0andB≥

wheredenotes the Peetre-type maximal function off,defined by

From this Peetre-type maximal function characterization forit follows easily that the spaceis independent of the choice ofφ0∈D(Rn)satisfying(1.1).

For anya∈(0,∞),s∈R,q∈(0,∞)andf∈the Lusin-area functionsandare defined,respectively,by setting,for allx∈Rn,

and

whereare as in Definition 1.1.Applying the Peetre-type maximal function characterization ofin(1.2),we can conclude the following Lusin-area function characterization ofwhose proof is presented in Section 3.

Theorem 1.1Let w∈a∈(0,∞),s∈Rn,p∈(0,∞)and q∈(0,∞).Then,there exists a positive constant C such that,for all f∈

Motivated by[8],we introduce atoms of the spaceas follows.

Definition 1.2Let s∈R,p∈(0,1],q∈[p,∞)and w∈.A distribution a∈is called a(p,q,s)w-atom ofif the following hold:

(i)a is supported on a cube Q⊂Rncentered at cQand of side length(Q).

(ii)a≤[w(Q)

(iii)If|Q|<1,then for any g∈S(),a polynomial P of degree at most N:=maxn1−s,0and a smooth cutof ffunction∈S()such that ηQ=1on Q and=0outside2Q,,here and hereafter,2Q denotes the cube centered atand of side length2(Q).

Now we give the following atomic characterization of the Triebel-Lizorkin spaces,which follows from the Calder´on reproducing formula(see Lemma 2.3 below)and the Lusin-area function characterization ofin Theorem 1.1(see Section 4 for its proof).

Theorem 1.2Let s∈R,p∈(0,1],q∈[p,∞)and w∈(Rn).Then,f∈if and only if f=whereandare(p,q,s-atoms.Moreover,there exists a positive constant C such that,for all f∈

where the infimum is taken over all the decompositions of f as above.

Next,we show that functions incan be decomposed into finite linear combinations of(p,q,s)w-atoms with their coefficients belonging to.The proof is given in Section 5 by invoking some ideas from[17].

Theorem 1.3Let s∈R,p∈(0,1],q∈[p,∞)and w∈(Rn).Then,every f∈admits an atomic decomposition f=where N∈N,are(p,q,s)w-atoms such that eachand≤Cffor somepositive constant C independent of f and N.

Indeed,is dense in(see Lemma 6.1 below).Consequently,we can establish a boundedness criteria for(sub)linear operators from to some(quasi)-Banach spaces as in[17,26–27].Before going into details,we first recall the following notion(see,for example,[17,26–27]).

Definition 1.3(i)A quasi-Banach space B is a vector space endowed with a quasi-normBwhich is non-negative,non-degenerate(namely,fB=0if and only if f=0),homogeneous,and obeys the quasi-triangle inequality,namely,there exists a constant K≥1such that,for all f,g∈B,f+gB≤K[fB+gB].

(ii)Let r∈(0,1].A quasi-Banach space with the quasi-normis called an r-quasi-Banach space iff+gfor all f,g∈.

(iii)For any r-quasi-Banach spacewith r∈(0,1]and a linear space Y,an operator T from Y to is said to be-sublinear if,for all f,g∈Y and λ,ν∈C,

and(f)−T(g)≤T(f−g)

Applying Theorem 1.3 and the density property ofwe can establish a criterion for the boundedness of operators on(see Section 6 for its proof).

Theorem 1.4Let s∈R,p∈(0,1],q∈[p,∞)and w∈.Suppose thatis an r-quasi-Banach space with r∈[p,1]and that T:→is a-sublinear operator satisfying that

Then T uniquely extends to a bounded Br-sublinear operator from

From Theorem 1.4,it follows the boundedness of the local Riesz operator on the Triebel-Lizorkin spaces with local weights.Let Φ∈D()satisfying Φ(x)=1 forx∈B(0,1)and suppΦ⊂B(0,2).Forj∈{1,···,n},consider the local Riesz operator

(see[22,25]).It was proved in[22,Lemma 8.2]thatis bounded onwhenp∈(1,∞)andw∈and fromtowhenw∈Forp∈(0,1],letbe the weighted local Hardy space,which consists of allf∈such that+:=∗f|∈whereφ∈has a non-zero integral,and we definef:=The operatorswere used to characterizein[22].Moreover,it was proved in[25,Theorem 8.2]thatis bounded on the Orlicz-type local Hardy spaces with local weights,which particularly implies that eachis bounded on the local weighted Hardy spacewithp∈(0,1]andw∈

Applying Theorem 1.4,we obtain the following conclusion(see Section 7 for its proof).

Theorem 1.5Let s∈R,p∈(0,1],q∈[1,∞)and w∈Then,for all j∈{1,···,n},the operatoris bounded on

Remark 1.1(i)Forp∈(0,1],it was proved in[18,Theorem 2.25]that=with equivalent quasi-norms.Thus,for allp∈(0,1],if we takes=0 andq=2 in Theorem 1.5,then everyis bounded on the spaceifw∈

(ii)The result in(i)is slightly weaker than the aforementioned corollary of[25,Theorem 8.2],which says thatis bounded onfor allw∈The reasons for this are as follows.The size condition of an-atom in[25,Definition 3.4]can be given by anynorm withr∈[1,∞]∩(p,∞);meanwhile,any weightw∈implies thatw∈for somer∈(1,∞).However,the size condition of a(p,2,0)w-atom in Definition 1.2 is given by some fixed quasi-normso we are forced to use weightsw∈

This article is organized as follows.In Section 2,we first recall some known basic lemmas,including the properties of the local weight,Fefferman-Stein vector-valued inequalities associated to the local weights,and the Caldern reproducing formulae;we then prove a duality result related to the spaceThe proof of Theorem 1.1 is presented in Section 3.The whole Section 4 focuses on the proof of Theorem 1.2,by using Theorem 1.1 and a series of auxiliary lemmas developed in Section 4.In Section 5,we prove Theorem 1.3 by using the atomic decomposition result in Theorem 1.2.Sections 6 is devoted to the proof of Theorem 1.4 by showing the density property ofin(see Lemma 6.1).Finally,the proof of Theorem 1.5 is presented in Section 7,by using Theorem 1.5.

Throughout this paper,we use the following notation.Let N:={1,···},Z+:=N∪{0}and Z:={0,±1,···}.Denote byCa positive constant independent of the main parameters involved,which may vary at different occurrences.We usefgorgfto denotef≤Cgorg≥Cf,respectively.Iffgf,then we writef∼g.

2 Preliminaries

Forκ∈(0,∞),the local Hardy-Littlewood maximal operatoris defined by setting,for all locally integrable functionsfandx∈Rn,

Ifκ=1,then we simply writeasAlso,forB≥0,all suitable functionsfandx∈Rn,letKB(f)(x):=|f(y)|e−B|x−y|dy.

The following versions of the vector-valued Fefferman-Stein maximal inequalities associated to local weights were proved in[18,Lemma 2.11].

Lemma 2.1If κ∈(0,∞),p∈(1,∞),q∈(1,∞]and w∈then,for any sequence of locally integrable functionsit holds true that

and there exists a positive constant B0,depending only on n and w,such that,when B≥,

where C is a positive constant depending only on n,κ,p,q,B and[w]

Some properties of the local weights are presented in the following lemma;whose proofs were given in[18,Lemma 1.4]and[22,Lemma 2.1 and Corollary 2.1].

Lemma 2.2Let p∈[1,∞],w∈and κ∈(0,∞).

(i)There exists a positive constant cw,which depends only on[wand n,such that,for all t∈[1,∞)and cubes Q with|Q|=1,w(tQ)≤

(ii)There is a positive constant C,which depends only onand n,such that,if|Q|≤1,then w(2Q)≤Cw(Q)and,if|rQ|>1,then w((r+1)Q)≤Cw(rQ).

(iii)is bounded fromif p=1,and bounded onif p∈(1,∞].

(iv)For p∈(1,∞),w∈if and only ifhere and hereafter,denotes the conjugate index of p.

The Caldern-type reproducing formula in the local case was essentially given in[18,Theorem 1.6].Indeed,Lemma 2.3 forj=0 was proved in[18,Theorem 1.6]and the proofs for the general casej∈N are essentially the same.We omit the details.

Lemma 2.3Assume thathas a nonzero integral.Let φ:=Then,for any given L∈Z+,there exist functionssuch that ψ0has a nonzero integral,ψ has vanishing moments up to order L(namely,dx=0for all multiindices α with|α|≤L)and,for all j∈Z+and f∈

where φi:=and ψi:=for all i∈N.

Finally,we conclude this section with the following duality result.

Proposition 2.1Let s∈R,p∈(1,∞),q∈[1,∞)and w∈Then

ProofForp∈(1,∞),q∈[1,∞)andw∈we denote bythe space of all sequences of functionssuch that:=is fi nite.Ifw=1,then we simply writeasBy an argument similar to that used in the proof of(see[24,p.177]),we conclude that

Letf∈For allg∈by Lemma 2.3,we haveg=g,where,are as in Lemma 2.3.Without loss of generality,we may assume thathas vanishing moments up to orderM>|s|.From this and Holder’s inequality,it follows that

with a usual modification made whenq=1.Thus,Lf(g):=induces a linear continuous functional on(Rn)withLf≤fHence,(Rn).

To show the converse,we assume thatL∈Since

is a one-to-one map fromto a subspace ofit follows that the functionalLcan be interpreted as a functional on that subspace ofBy the Hahn-Banachtheorem,Lcan be extended to a continuous linear functional onwith the norm preserved,which is denoted byBy this and=(Rn),we see that there existssuch that,for allf∈(Rn),

where(−·)for allj∈Z+and==If we letg:=thenL(f)=g(x)f(x)dxfor allf∈Sincehave compact support and vanishing moments up to orderM>|s|,it follows that,for allx∈Rn,

withκbeing a sufficiently large number depending on the support ofThen,

By Hlder’s inequality and1,we see that the last term displayed above is bounded by

Finally,we apply(2.1)and Lemma 2.2(iv)to obtain

Hence,(Rn),which completes the proof of Proposition 2.1.

3 Proof of Theorem 1.1

In this section,we show Theorem 1.1 by using the following estimate in[18,Lemma 2.9].

Lemma 3.1Assume that φ0∈D(Rn)has a nonzero integral.Let φ:=φ0−Then,for any r∈(0,∞),A≥0and B≥0,there exists a positive constant C,depending only on n,r,φ0,A and B,such that,for all f∈,j≥0and x∈Rn,

Proof of Theorem 1.1Since(f)(x)≤(f)(x)for allx∈Rn,it follows that

For allA,B∈(0,∞),x∈andk∈N,by the definition ofwe obtain

Consequently,

which,combined with(1.2),implies that

It remains to show thatTo this end,we chooser∈(0,min{p,q}),A>maxandB>(withB0as in Lemma 2.1).Then,from Lemma 3.1,we deduce that,for allj∈Z+andx∈Rn,

Combining this with Fubini’s theorem and the fact that,whenk≥j≥0 and|z|≤,

we further conclude that,for allx∈Rn,

Let

Further,since 00,it follows,from Holder’s inequality,that

ByAr>n,we see that,for allj∈Z+andx∈Rn,

which implies that

Consequently,we have

Further,byr

4 Proof of Theorem 1.2

To prove Theorem 1.2,we need to establish a series of auxiliary lemmas.

Lemma 4.1Let s∈(−∞,0),p∈(0,1],q∈[p,∞)and w∈Assume that

(i)φ0∈D(Rn)such thatsupp(φ0)⊂B(0,1);

(ii)for any z∈Rn,φ(z):=φ0(z)−

(iii)a∈is a(p,q,s)w-atom supported on a cube Q,with center cQand side length

Then,for all k∈Z+,supp(φk∗a)⊂B(cQ,3).Moreover,for all k∈Z+and x∈B(cQ,3),

where C is a positive constant independent of k,a and x.

ProofIt is easy to see that every supp(φk∗a)⊂B(cQ,3).Since(Q)≤it follows that there existsi0∈Z+such thatBy Definition 1.2,the(p,q,s)w-atomahas vanishing moments up to orderN:=−s.From the Calder´on reproducing formula(2.3),it follows that there exist functionsψ0,ψ∈D(Rn)such thatψ0has a nonzero integral andψhas vanishing moments up to orderN,and

Letbe the Taylor polynomial given by that,for ally,z∈Rn,

whereare coefficients.For anyy,z∈Rn,set:=(z−y)−(z;y).LetηQbe the smooth cutof ffunction associated to the cubeQas defined in Definition 1.2.For allk∈Z+andz∈Rn,by the vanishing moment condition ofa,we have∗a(z)==From this and(4.1),it follows that,for allk∈Z+andz∈Rn,

where we used the notation(u):=ϕ(−u)for any functionϕandu∈Rn.By the choice ofi0and the support conditions ofφanda,we conclude that supp⊂Band supp(∗a)⊂B(,fori>i0.Thus,for allk∈Z+andz∈Rn,

For all multi-indicesα∈with|α|≤N,the mean value theorem further implies that,for ally∈B(cQ,andz∈Rn,

From(4.3),it further follows that

By(4.3)and(Q)∼,we know that,for ally∈B

Sincedx=0 for all|α|≤N,we see that,for alli≥i0+1 andy∈B

By the mean value theorem and(4.4),we see that the quality inside the bracket of the second line of(4.6)is dominated by

Inserting this into(4.6)gives that,wheni>i0,

Applying(4.2),(4.5)and(4.7),we conclude that,for allk∈Z+andz∈Rn,

Now we consider the following two cases.

Case 1q∈[1,∞).Takingz:=x∈B(cQ,3)in(4.8)and applying the factwe see that

Further,if we apply Hlder’s inequality to each term on the right-hand side of the above inequality,and use the factss<0 ands+N+1>0,then

Using Hlder’s inequality andw∈we further have

Inserting this estimate into(4.9),we conclude that,for allx∈B(cQ,3),

which is the desired conclusion.

Case 2p∈(0,1)andq∈[p,1).In this case,w∈and−sSincew∈it follows that the ballBis covered by a finite number of(depending only onn)smaller cubes{}such that eachhas the same side length as that ofQand hencew()∼w(Q)by Lemma 2.2(ii),which further gives that

From(4.8),(4.10)and the fact=together withs<0,we deduce that,for allk∈Z+andz∈Rn,

which,together withn+s+N+1>0 andq<1,gives that

Sinceq≥pandN+1>−s,we have−n≥(n+s+N+1)p−n>0 and hence,for alli≤by(Q).Combining this withB(cQ,⊂B(cQ,3),we further know that,for allk∈Z+andz∈Rn,

Let A:=φk∗a(z)|and(a)be the Peetre-type maximal function as defined in(1.3),whereA>A0andB>By the facts thatw∈andn+N+1+s>0,together witha∈and(1.2),we see that

Since A is finite,by(4.11),we obtain

that is,A[w(Q)From this and the definition of A,we deduce that,for allk∈Z+andx∈B(cQ,3),

which completes the proof of Lemma 4.1.

Lemma 4.2If s∈R,p,q∈(0,∞)and w∈then there exists a positive constantC such that,for all f∈

where Dj:=for j∈{1,···,n}.

ProofFixp,q∈(0,∞)andw∈Form∈Z+,it was proved by Rychkov[18,Theorem 2.20]that,for allf∈

where,forα:=

The proof for(4.12)is an easy adaption of(4.13)and the following lifting property in[18,Theorem 2.18]:There existst∈(0,∞)sufficiently small,depending onp,nandw,such that,for alla∈R andf∈

wheredenotes the Bessel potential operatorwhere id is the identity operator.Precisely,for anys∈R,we choosem∈Z such thatm≤|s|

which implies(4.12).This finishes the proof of the lemma.

Lemma 4.3Let s∈R,p∈(0,1],q∈[p,∞)and w∈Then,there exists a positive constant C,depending only on s,p,q,n and w,such that,for all(p,q,s)w-atoms a,

ProofLetabe a(p,q,s)w-atom supported on a cubeQwith centercQand side length(Q).Ifp=q,then(4.15)follows trivially from Definition 1.2(ii),so it suffices to show(4.15)forp

Case 1ands<0.For anyk∈Z+,we observe that supp(φk)⊂Bwhich gives that supp⊂Qby using suppa⊂Q.Here and in what follows,we useQ(x,r)to denote the cube in Rnwith centerx∈Rnand side lengthr∈(0,∞).From this and Holder’s inequality,together with Lemma 2.2(ii),it follows that

This proves(4.15)for the case

Case 2Q)ands<0.In this case,for anyk∈Z+,by supp(φk)⊂Band suppa⊂Q,we have supp(φk∗a)⊂B(cQ,2+⊂B(cQ,3),which implies that

Following the argument used in Case 1,we apply Hlder’s inequality andw(4to deduce that Z11.

Now westimate Z2.If x ∈ B(cQ,3)such that0,then byφk∗a(x)=we have+which implies that|x−that is,k≤By this and Lemma 4.1,we see that

whereκis a positive constant independent ofaandx.Since>,by the weighted boundedness properties ofMκwe further know that

Combining the estimates of Z1and Z2,we see that(4.15)holds whenQ)ands<0.

Based on the conclusions in Cases 1–2,we know that(4.15)holds whens<0.The general case of(4.15)can be reduced to the cases<0 by using Lemma 4.2.Indeed,if 0≤s<1 andais a(p,q,s)w-atom,then by Lemma 4.2,we see that

If we can show thataandare constant multiples of(p,q,s−1)w-atoms,then applying the already proved conclusion that(4.15)holds fors<0 yields thata1 and1,which,combined with(4.16),gives thata1,that is,(4.15)holds for all 0≤s<1.Repeating the above process yields(4.15)for alls∈R.

To finish the proof of Lemma 4.3,we still need to show that ifais a(p,q,s)w-atom,thenaandare constant multiples of(p,q,s−1)w-atoms.It is obvious,from Definition 1.1,that≤and henceais a(p,q,s−1)w-atom.Likewise,everyDjais a constant multiple of a(p,q,s−1)w-atom provided that we can prove that

To show(4.17),we first shift the differential fromatoφ,namely,andfori∈N,and then use the Calder´on reproducing formula(see Lemma 2.3)and the fact that,for allk,i∈Z+andx∈Rn,

whereL∈(0,∞)can be sufficiently large andCis a positive constant depending on the supports ofφandψ.Based on these facts,(4.17)follows from a standard calculation(see the proof of[18,Theorem 2.5])and we thus omit the details.

Summarizing all the above arguments,we complete the proof of Lemma 4.3.

Proof of Theorem 1.2Fixs∈R,p∈(0,1],q∈[p,∞)andw∈If there exist(p,q,s)w-atomssuch thatf=then we apply Lemma 4.3 to obtain This inequality implies thatf,where the infimum is taken over all the atomic decompositions offas above.

To show the converse part,fixf∈Assume thathas a nonzero integral,0≤φ0≤1,supp(φ0)=1 whenever|x|≤,and thatφ:=satisfies(1.1).For allx∈Rn,define

Then,by Theorem 1.1,we have<∞.Denote byQthe collection of all dyadic cubes of Rn.For anyk∈Z,let Ωk:=and

Notice that Λkmight be empty.It is easy to see that,for eachQ∈Qwith|Q|≤1,there exists a uniquek∈Z such thatQ∈Λk.A dyadic cubeQ∈Λkis called a maximal dyadic cube in Λkif,for any dyadic cube∈Λk,either⊂Qor∩Q= ∅.For eachk∈N,denote bythe collection of all such maximal dyadic cubes in,whereis the index set(which might be empty).Observe thatare mutually disjoint.Moreover,

Given anyL∈Z+sufficiently large(to be determined later),by Lemma 2.3,we see that

wherehas a nonzero integral andψ∈D(Rn)satisfies thatdx=0 for all multi-indicesαwith|α|≤L.Without loss of generality,we may assume that bothandare supported onBfor all∈Z+.For eachQ∈Qwith|Q|≤1,there exists somesuch that(Q)=,and then we define

With these notations,we rewrite(4.19)as

For eachk∈Z andi∈Ik,let

and,for allx∈Rn.

Based on(4.21),we see thatf=is the desired atomic decomposition off,provided that we can show that everyis a(p,q,s)w-atom and

To prove(4.22),we first show that,for allk∈Z,

To prove(4.23),let(x)>for allk∈Z.From(2.1)and Lemma 2.2(iii),it follows thatand hence

On the other hand,for ally∈Q∈,we haveQ⊂and

which further implies that

Combining(4.24)–(4.25),we obtain(4.23).

From Hlder’s inequality,the fact that the familyare mutually disjoint,and(4.23),we deduce that

which shows(4.22).

Now we show that eachis a constant multiple of some(p,q,s)w-atom.Indeed,by supp(φ0)⊂B(0,1),it is easy to see that suppAlso,notice thatsatisfies the vanishing moment condition provided that<1.Thus,it remains to prove that This can be done by considering the following two cases:q∈(1,∞)andq∈[p,1].

Case 1q∈(1,∞).Byw∈we see that,for allQ∈Qwith|Q|≤1,

From Theorem 1.1 and the fact thatQ:|x−y|

Applying(4.26)–(4.27)and Hlder’s inequality,we conclude that,for allg∈(Rn)with a norm at most 1,From this and Proposition 2.1,it follows that

Case 2q∈[p,1].In this case,notice thatw∈By Theorem 1.1,we have

For allx∈Pand|x−z|<(P),byq∈[p,1]and the inequality thatholds for all sequenceswe conclude that

Becauseφandψsatisfy the vanishing moment condition up to orderL,we apply[6,Lemma 2 in p.121 and Lemma 4 in p.122]or[17,Corollary 3.1]to deduce that,for allu∈Rn,

Combining this with supp⊂{u:|u|≤2max{P),(Q)}}further implies that,for ally∈Q,x∈Pand|x−z|<(P),if(z−y)0,then

and

Invoking(4.30)–(4.31),we continue to estimate(4.29)with

Inserting this into(4.28)and interchanging the summations inPandQ,we obtain

which gives,provided that whenQ⊂

To show(4.32),we define

whereW1:={P∈Q:(P)≤Q),|P|≤1,{(P),(Q)}}and

Notice that,for allP∈W1,we have≤1 andP⊂so that1 by Lemma 2.2(ii).Therefore,if we chooseL>|s|,then

Now we estimate Z2.Observe thatQ⊂5whenP∈W2.Since(P)≤1 and(Q)≤1,the factw∈implies thatw(x).Thus,

By this and the expression ofwe see that,whenLsatisfies(L+s)q+n(q−1)>0,

Combining(4.33)–(4.34)gives(4.32).Thus,we complete the proof of Theorem 1.2.

5 Proof of Theorem 1.3

Proof of Theorem 1.3Letf∈With all the notation as in the proof of Theorem 1.2,we decomposefintowhere eachis a(p,q,s)w-atom supported ondefined by

andHere,for anyk∈Z,is as in(4.18);is the largest dyadic cube contained in the setthe functionsandare as in(4.20).Without loss of generality,we may assume thatwithj∈are supported onB(0,),andwithj∈N have the vanishing moments up to orderL,whereL>|s|.

For anyN∈N,let

For eachQ∈there exists a unique(k,i)such thatQ⊂Denote byJNthe collection of all such(k,i).Sincehas finitely many elements,so doesJN.For each(k,i)∈JN,let

In a similar way to the arguments used in Cases 1–2 of the proof of Theorem 1.2,we conclude that eachis also a constant multiple of some(p,q,s)w-atom.Sinceand the summation in(5.1)has only finite terms,it follows that everyFor anyN∈N,setfN:=andThen,bothfNandbNare inMoreover,fNis a linear combination of finite smooth(p,q,s)w-atoms,and the proof of Theorem 1.2 shows that thep-(quasi)norm of its coefficients is bounded by a constant multiple ofThus,to finish the proof of Theorem 1.3,we only need to prove that,whenNis large enough,bNis a small constant multiple of some(p,q,s)w-atom.

For allx∈Rn,noticing thatwe write

Then,applying(4.21),we see that,for allx∈Rn,

Sincefhas compact support,there existsN0∈N such that suppf⊂Q(0,).Notice that supp(φQ)⊂Q.For anyto ensure that∗fis a non-zero function,we needQ∩But any cubeQinis dyadic and satisfies(Q)≤1,and we therefore conclude thatQ∩Qis the same asQ⊂Notice that,ifN>N0+5,then{Q∈:Q⊂={Q∈Q:(Q)<}.This allows us to replace the summation in(5.2)withwhenN>N0+5.Thus,whenN>N0+5,we have

whereandare associated toandas in(4.20).Clearly,supp(bN)⊂Now we estimate the quasi-normFrom(5.3),it follows that

If0,then by the support conditions ofφjandψwe see that|j−|≤2.Thus,

For the sake of simplicity,we only estimate the termj=By supp⊂and the estimate that,for allϕ,g∈S(Rn)such that=0 for all multi-indices|α|≤L,

(see[17,Lemma 3.3(i)]or[6,p.121,Lemma 2]),we have

which is bounded by a constant multiple ofprovided thatNis large enough(depending onw,N0,p,q,sandL).Therefore,

for some positive constantcdepending only onw,N0,n,p,q,Lands,and

is a(p,q,s)w-atom supported onby observing thataNdoes not need to satisfy any vanishing moment since≥1,which implies that

is a finite atomic decomposition with(p,q,s)w-atomsand the coefficients satisfying

This finishes the proof of Theorem 1.3.

6 Proof of Theorem 1.4

The goal of this section is to show Theorem 1.4.We need the following density lemma.

Lemma 6.1Let s∈R,p,q∈(0,∞)and w∈is dense in

ProofBy the localization principle in[18,Theorem 2.21],for allf∈we have

whereγ∈D(Rn)such that its integer translatesfor allx∈Rnwithk∈Znform a partition of unity,that is,=1 for allx∈Rn.Notice that supp(γk)has finite overlapping property(bounded by a positive constant depending only on suppγ).For allN∈N,definefN:=Observe that everyfNhas compact support.Moreover,the sequenceconverges tofasN→∞.Indeed,by(6.1),we see that,whenN→∞,

wherecis a positive constant depending only on suppγ.Thus,to finish the proof of this lemma,we may as well assume thatf∈has compact support.

Now suppose thatf∈has compact support.By(2.3),writefwithψiandφias in Lemma 2.3.Without loss of generality,we may assume that bothψiandφiare supported on the ballB(0,2−i)fori∈For allN∈N,letgN:=Sincefis assumed to have compact support,it follows that everygNIt remains to show thatgN→finasN→∞.Notice that

For allx∈Rn,from the previous assumptions that bothandare supported on the ballB(0,),the definition of the Peetre-type maximal function,and[17,Corollary 3.1](see also[6,Lemma 2 in p.121,and Lemma 4 in p.122]),we deduce that,for allx∈Rn,

whereAandBare positive constants satisfyingA≥A0andBandM>A+|s|is a sufficiently large constant.By this and Holder’s inequality,we further conclude that

which tends to 0 asN→∞in terms of(1.2).Thus,and it converges tofinasN→∞,which completes the proof of Lemma 6.1.

Proof of Theorem 1.4For anyf∈by Theorem 1.3,fadmits a finite atomic decompositionf=whereN∈N,are(p,q,s)w-atoms inandBy the assumption of Theorem 1.4,we see that1 for allk∈{1,···,N}.Moreover,byr∈[p,1]and the fact thatTisBr-sublinear,we have

In general,for anyf∈by Lemma 6.1,there existsthat converges tofinasm→∞.By(6.2)and Definition 1.3(iii),is a Cauchy sequence inBrand hence it converges to some element inBr,which we denote byf,namely,f:=inBr.Notice thatis well defined based on(6.2)and Definition 1.3(iii).Consequently,=,which completes the proof of Theorem 1.4.

7 Proof of Theorem 1.5

In this section,we apply Theorem 1.4 to show Theorem 1.5.

Proof of Theorem 1.5Notice thatforp∈(0,1]andq∈[p,∞),is ap-quasi-Banach space.By Theorem 1.4,it suffices to show that,for any(p,q,s)w-atomasupported on a cubeQ,

To this end,by Theorem 1.1,we only need to show that1,where

Heresatisfies the vanishing moments up to orderL≥max{−1,},andfork∈N.Without loss of generality,we may assume that every supp(φk)⊂B(0,2−k)andL>|s|.

By the support conditions ofaand,together with the definition ofwe see that supp<(Q)+3}.Applying Hlder’s inequality with=1 and Lemma 2.2(ii),we know that

By the Caldern reproducing formulae and the fact thatcommutates with the convolution operator,we conclude that,for ally∈Rn,

whereandare as in Lemma 2.3(here we also assume thatandare supported onhave vanishing moments up to orderL>|s|).Furthermore,

where we used again[6,Lemma 2 in p.121 and Lemma 4 in p.122](see also[17,Corollary 3.1]).Asq>1,using Holder’s inequality and the definition ofgives that,when|x−y|<

Notice thatLis taken to be larger than|s|.Then,applying Holder’s inequality again,we obtain

By this and Fubini’s theorem,together with the fact that

we see that

Then,using the fact thatand the size condition of a(p,q,s)w-atom,we see that the last quality in the above estimate is bounded by

Thus,1.Hence,(7.1)holds and we complete the proof of the theorem.

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