# Herz-Morrey-Hardy Spaces with Variable Exponents and Their Applications

Authors: Jingshi Xu; Xiaodi Yang
Journal: Journal of Function Spaces
 (2015)

---

## Abstract

The authors introduce Herz-Morrey-Hardy spaces with variable exponents and establish the characterization of these spaces in terms of atom. Applying the characterization, the authors obtain the boundedness of some singular integral operators on these spaces.


---

## Body

## 1. Introduction

The Herz spaces go back to Beurling and Herz; see [1, 2]. Firstly, they attracted a lot of authors’ attention because they could be used to characterize Fourier multipliers for Hardy spaces; see [3]. Then, in 1989 Chen and Lau in [4] and García-Cuerva in [5] introduced now called nonhomogeuous Herz type Hardy spaces. They found that these Herz type Hardy spaces have a decomposition via central atoms. After that, Lu et al. considered homogeuous Herz type Hardy spaces and also obtained a central atomic decomposition for them. Since then Herz type spaces have been studied extensively; see monograph [6] for details. Meanwhile, in the last three decades, the interest of the study for variable exponent spaces has been increasing year by year. Variable exponent spaces have many applications: in electrorheological fluid [7], in differential equations [8] and references therein, and in image restoration [9–11], for instance. Indeed, many spaces with variable exponents appeared, such as: Lebesgue spaces, Sobolev spaces and Bessel potential spaces with variable exponent, Besov and Triebel-Lizorkin spaces with variable exponents, Morrey spaces with variable exponents, Campanato spaces with variable exponent, and Hardy spaces with variable exponent; see [12–23] and references therein. Moreover, the atomic, molecular, and wavelet decompositions of variable exponent Besov and Triebel-Lizorkin spaces were given in [13, 14, 20, 21, 24]. The duality and reflexivity of spaces Bp(·),qs and Fp(·),qs were discussed in [25]. The atomic and molecular decompositions of Hardy spaces with variable exponent and their applications for the boundedness of singular integral operators were obtained in [22, 26].Recently, as a generalization of Lebesgue spaces with variable exponent, Herz spaces with variable exponents are introduced. In fact, in 2010 Izuki proved the boundedness of sublinear operators on Herz space with variable exponentsK˙p(·)α,q and Kp(·)α,q in [27]. In 2012, Almeida and Drihem obtained boundedness results for a wide class of classical operators on Herz spaces K˙p(·)α(·),q and Kp(·)α(·),q in [28]. Shi and the first author in [29] considered Herz type Besov and Triebel-Lizorkin spaces with one variable exponent. Then Dong and first author in [30] established the boundedness of vector-valued Hardy-Littlewood maximal operator in spaces K˙p(·)α(·),q and Kp(·)α(·),q and gave characterizations of Herz type Besov and Triebel-Lizorkin spaces with variable exponents by maximal functions. In [31], Wang and Liu introduced a certain Herz type Hardy spaces with variable exponent. In 2013, Samko introduced Herz spaces with three variable exponents and obtained the boundedness of Hardy-Littlewood maximal operator on them. In [32–34], the boundedness of singular integrals and their commutators of BMO functions are discussed in Herz Morrey spaces with variable exponents. The Herz-Morrey spaces with constants were considered in [35, 36]; however, there is no theory of Herz-Morrey type Hardy spaces. In this paper we fill the gap and introduce Herz-Morrey-Hardy spaces with variable exponents.The outline of the paper is as follows. In the rest of the section we will recall some definitions and notions. In Section2, we will define the Herz-Morrey-Hardy spaces with variable exponents HMK˙p(·),λα(·),q and HMKp(·),λα(·),q and give their atomic characterization. In Section 3, we obtain that certain singular integral operators are bounded from Herz-Morrey-Hardy spaces with variable exponents into Herz-Morrey spaces with variable exponents as an application of the atomic characterization.Throughout this paperE denotes the Lebesgue measure and χE the characteristic function for a measurable set E⊂Rn. For a multi-index β=(β1,β2,…,βn), we denote |β|=β1+β2+⋯+βn. We also use the notation a≲b if there exists a constant c>0 such that a⩽cb. If a≲b and b≲a we will write a≈b. Finally we claim that C is always a positive constant but it may change from line to line.Definition 1.
LetE be a measurable set in Rn with |E|>0. Let p(·):E→[1,∞) be a measurable function. Denote(1)Lp·E:=fismeasurableon⁡E:ρp·fλ<∞fλforsomeconstantλ>0,where ρp(·)(f):=∫E‍|f(x)|p(x)dx, and (2)fLp(·)(E):=inf⁡λ>0:ρp·fλ⩽1.Then Lp(·)(E) is a Banach space with the norm ∥·∥Lp(·)(E).LetLloc⁡1(Rn) be the collection of all locally integrable functions on Rn. Given a function f∈Lloc⁡1(Rn), the Hardy-Littlewood maximal operator M is defined by (3)Mf(x):=sup⁡r>0r-n∫Bx,r‍fydy,∀x∈Rn,where and what follows B(x,r):={y∈Rn:|x-y|<r}. We also use the following notation: p-:=essinf⁡{p(x):x∈Rn} and p+:=esssup⁡{p(x):x∈Rn}. The set P(Rn) consists of all p(·) satisfying p->1 and p+<∞. B(Rn) is the set of p(·)∈P(Rn) satisfying the condition that M is bounded on Lp(·)(Rn). It is well known that if p(·)∈P(Rn) satisfies the following global log-Hölder continuous then p(·)∈B(Rn); see [37–42].Definition 2.
Letα(·) be a real-valued function on Rn. If there exists C>0 such that, for all x,y∈Rn, |x-y|<1/2,(4)αx-αy⩽C-log⁡x-y,then α(·) is said local log-Hölder continuous on Rn.
If there existsC>0, such that for all x∈Rn,(5)αx-α0⩽Clog⁡(e+1/x),then α(·) is said log-Hölder continuous at origin.
If there existα∞∈R and a constant C>0 such that for all x∈Rn(6)αx-α∞⩽Clog⁡(e+x),then α(·) is said log-Hölder continuous at infinity.
Ifα(·) is both local log-Hölder continuous and log-Hölder continuous at infinity, then α(·) is said global log-Hölder continuous.The sets of log-Hölder continuous functions, log-Hölder continuous functions at origin, log-Hölder continuous functions at infinity, global log-Hölder continuous are denoted byPloc⁡log⁡(Rn), P0log⁡(Rn), P∞log⁡(Rn), and Plog⁡(Rn), respectively.We denotep′(·) by the conjugate exponent to p(·), which means p′(·)=p(·)/(p(·)-1). It is also well known that p(·)∈B(Rn) is equivalent to p′(·)∈B(Rn); see [39].For simplicity, we denoteLp(·)(Rn) by Lp(·). We will use the following results.Lemma 3 (see [43]).
Letp(·)∈P(Rn). If f∈Lp(·) and g∈Lp′(·), then fg is integrable on Rn and(7)∫Rn‍fxgxdx⩽rpfLp(·)gLp′(·),where rp=1+1/p--1/p+.Lemma 4 (see [27]).
Letp(·)∈B(Rn). Then there exist 0<δ1, δ2<1, and a positive constant C depending only on p(·) and n such that for all balls B in Rn and all measurable subsets S⊂B,(8)χBLp(·)χSLp(·)⩽CBS,χSLp·χBLp·⩽CSBδ1,χSLp′(·)χBLp′(·)⩽C|S||B|δ2.Lemma 5 (see [27]).
Letp(·)∈B(Rn). Then there exists a positive constant C such that, for any ball B in Rn,(9)χBLp(·)χBLp′(·)⩽CB.To give the definition of Herz-Morrey spaces with variable exponents, let us introduce the following notations. Letk∈Z, Bk:={x∈Rn:|x|⩽2k}, L∈Z, Dk:=Bk∖Bk-1, and χk:=χDk. The symbol N0 denotes the set of all nonnegative integers. For m∈N0, we denote χ~m:=χDm if m≥1 and χ~0:=χB0.Definition 6.
Let0<q⩽∞, p(·)∈P(Rn), and 0≤λ<∞. Let α(·) be a bounded real-valued measurable function on Rn. The homogeneous Herz-Morrey space MK˙p(·),λα(·),q and nonhomogeneous Herz-Morrey space MKp(·),λα(·),q are defined, respectively, by(10)MK˙p(·),λα(·),q:=f∈Lloc⁡p(·)(Rn∖0):fMK˙p(·),λα(·),q<∞,MKp(·),λα(·),q:=f∈Lloc⁡p(·)(Rn):fMKp(·),λα(·),q<∞,where(11)fMK˙p(·),λα(·),q:=sup⁡L∈Z2-Lλ∑k=-∞L‍2α(·)kfχkLp(·)q1/q,fMKp(·),λα(·),q:=sup⁡L∈N02-Lλ∑k=0L‍2α(·)kfχ~kLp(·)q1/q.Here there is the usual modification when q=∞.Proposition 7.
Letp(·)∈P(Rn), q∈(0,∞], and λ∈[0,∞). If α(·)∈L∞(Rn)∩P0log⁡(Rn)∩P∞log⁡(Rn), then(12)fMK˙p(·),λα(·),q≈max⁡sup⁡L⩽0,L∈Z2-Lλ∑k=-∞L‍2kα(0)qfχkLp(·)q1/q,sup⁡L>0,L∈Z2-Lλ∑k=-∞-1‍2kα(0)qfχkLp(·)q1/q+2-Lλ∑k=0L‍2kα(∞)qfχkLp(·)q1/q.Proposition7 is the generalization of Herz spaces with variable exponents in [28], and it was used in [33, 34].Lemma 8.
Letp(·)∈B(Rn), 0<q<∞, and λ∈[0,∞). Let α(·) be bounded and log-Hölder continuous both at the origin and at infinity such that -nδ1<α(0)⩽α∞<nδ2, where 0<δ1, δ2<1 are constants in Lemma 4. Suppose that T is a sublinear and bounded operator on Lp(·) satisfying size condition(13)Tfx⩽C∫Rn‍x-y-nfydyfor all f∈Lloc⁡1(Rn) with compact support and a.e. x∉suppf. Then there exists a positive constant C such that(14)TfMK˙p(·),λα(·),q⩽CfMK˙p·,λα·,q,TfMKp(·),λα(·),q⩽CfMKp(·),λα(·),qfor any function f belongs to MK˙p(·),λα(·),q and MKp(·),λα(·),q, respectively.Lemma8 is the generalization of Herz spaces with variable exponents in [28]. For a proof, see [33].


## 2. The Atomic Characterization

In this section, we will introduce Herz-Morrey-Hardy spaces with variable exponentsHMK˙p(·),λα(·),q and HMKp(·),λα(·),q. To do this, we need to recall some notations. S(Rn) denotes the Schwartz space of all rapidly decreasing infinitely differentiable functions on Rn, and S′(Rn) denotes the dual space of S(Rn). Let GNf be the grand maximal function of f defined by(15)GNf(x):=sup⁡ϕ∈ANϕ▿∗fx,x∈Rn,where AN:={ϕ∈S(Rn):sup⁡|α|,|β|⩽N,∀x∈Rn|xαDβϕ(x)|⩽1} and N>n+1 and ϕ▿∗ is the nontangential maximal operator defined by(16)ϕ▿∗fx:=sup⁡|y-x|<tϕt∗fy,∀x∈Rnwithϕt·=t-nϕ·t.The grand maximal operatorGN was firstly introduced by Fefferman and Stein in [44] to study classical Hardy spaces. For classical Hardy spaces, one can also see [45–47]. Nakai and Sawano generalized them to variable exponent case in [22].Definition 9.
Letα(·)∈L∞(Rn), 0<q≤∞, p(·)∈P(Rn), 0≤λ<∞, and N>n+1. The homogeneous Herz-Morrey-Hardy space with variable exponents HMK˙p(·),λα(·),q and nonhomogeneous Herz-Morrey-Hardy space with variable exponents HMKp(·),λα(·),q are defined, respectively, by(17)HMK˙p(·),λα(·),q:=f∈S′(Rn):fHMK˙p(·),λα(·),q:=GNfMK˙p(·),λα(·),q<∞,HMKp(·),λα(·),q:=f∈S′(Rn):fHMKp(·),λα(·),q:=GNfMKp(·),λα(·),q<∞.Remark 10.
Ifα(·)≡α and λ=0, these spaces were considered by Wang and Liu in [31]. If p(·) and α(·) are constant and λ=0, these are the classical Herz type Hardy spaces; see [6].Letψ(r)=1 for r∈[0,1] and ψ(r)=r-N for r∈(1,+∞). Then there exists C>0 such that ϕ(x)≤Cψ(|x|) for all ϕ∈AN. Therefore, by [46, Proposition in Page 57], there exists C>0 such that GNf(x)⩽CMf(x) for all x∈Rn. This means that GNf satisfies the size condition in Lemma 8. By Lemma 8, if -nδ1<α(0)⩽α∞<nδ2 and p(·)∈B(Rn), then(18)HMK˙p(·),λα(·),q∩Lloc⁡p·Rn∖0=MK˙p·,λα·,q,HMKp(·),λα(·),q∩Lloc⁡p(·)(Rn)=MKp(·),λα(·),q.Thus we are interested in the casenδ2⩽α(0), α∞<∞. In this case, we will establish a characterization of the spaces HMK˙p(·),λα(·),q and HMKp(·),λα(·),q in terms of central atom. For u∈R we denote by [u] the largest integer less than or equal to u.Definition 11.
Letp(·)∈P(Rn) and α(·)∈L∞(Rn) be log-Hölder continuous both at the origin and infinity, and nonnegative integer s⩾[αr-nδ2]; here αr=α(0), if r<1, and αr=α∞, if r⩾1, nδ2⩽αr<∞ and δ2 as in Lemma 4.(i)
A functiona on Rn is called a central (α(·),p(·))-atom, if it satisfies (1) supp a⊂B(0,r); (2) aLp(·)⩽|B(0,r)|-αr/n; (3) ∫Rn‍a(x)xβdx=0, β⩽s.(ii)
A functiona on Rn is called a central (α(·),p(·))-atom of restricted type, if it satisfies (1) supp a⊂B(0,r), r⩾1; (2) aLp(·)⩽|B(0,r)|-αr/n; (3) ∫Rn‍a(x)xβdx=0, β⩽s.Remark 12.
Ifp(·)≡p and α(·)≡α are constant, then taking δ2=1-1/p we recover the classical case in [6].Theorem 13.
Let0<q<∞, p(·)∈B(Rn), 0≤λ<∞, and α(·)∈L∞(Rn) be log-Hölder continuous both at the origin and infinity, 2λ⩽α(·), nδ2⩽α(0), α∞<∞, and δ2 as in Lemma 4.(i)
f∈HMK˙p(·),λα(·),q if and only if f=∑k=-∞∞‍λkak in the sense of S′(Rn), where each ak is a central (α(·),p(·))-atom with support contained in Bk and sup⁡L∈Z2-Lλ∑k=-∞L‍|λk|q<∞. Moreover,(19)fHMK˙p(·),λα(·),q≈inf⁡sup⁡L∈Z2-Lλ∑k=-∞L‍|λk|q1/q,where the infimum is taken over all above decompositions off.(ii)
f∈HMKp(·),λα(·),q if and only if f=∑k=0∞‍λkak in the sense ofS′(Rn), where each ak is a central (α(·),p(·))-atom of restricted type with support contained in Bk and sup⁡L∈Z2-Lλ∑k=0L‍|λk|q<∞. Moreover(20)fHMKp(·),λα(·),q≈inf⁡sup⁡L∈Z2-Lλ∑k=0L‍λkq1/q,where the infimum is taken over all above decompositions off.Proof.
We only prove (i). The proof of (ii) is similar. We use the ideas in [6]. To prove the necessity, we choose ϕ∈C0∞(Rn) such that ϕ⩾0, ∫Rn‍ϕ(x)dx=1, and suppϕ⊂{x:|x|⩽1}. For j∈N0, let ϕ(j)(x):=2jnϕ(2jx), ∀x∈Rn. For each f∈S′(Rn), set f(j)(x)=f∗ϕ(j)(x), ∀x∈Rn. It is obvious that f(j)∈C∞(Rn) and limj→∞f(j)=f in S′(Rn). Let ψ be a radial smooth function such that suppψ⊂{x:1/2-ɛ⩽|x|⩽1+ɛ} with 0<ɛ<1/4, ψ(x)=1 for 1/2⩽|x|⩽1. Let ψk(x):=ψ(2-kx) for k∈Z and (21)A~k,ɛ:={x:2k-1-2kɛ⩽x⩽2k+2kɛ}.Observe that suppψk⊂A~k,ɛ and ψk(x)=1 for x∈Ak:={x:2k-1⩽|x|⩽2k}. Obviously, 1⩽∑k=-∞∞‍ψk(x)⩽2, x≠0. Let(22)Φk(x):=ψkx∑l=-∞∞‍ψlx,x≠00,x=0.Then ∑k=-∞∞‍Φk(x)=1 for x≠0. For each m∈N, we denote by Pm the class of all the real polynomials with the degree less than m. Let Pk(j)(x):=PA~k,ɛ(f(j)Φk)(x)χA~k,ɛ∈Pm(Rn) be the unique polynomial satisfying(23)∫A~k,ɛ‍f(j)(x)Φk(x)-Pk(j)(x)xβdx=0,|β|⩽m=max⁡α0-nδ2],[α∞-nδ2.Write(24)fjx=∑k=-∞∞‍f(j)(x)Φk(x)-Pk(j)(x)+∑k=-∞∞‍Pk(j)(x):=I(j)+II(j).For the term I(j), let gk(j)(x):=f(j)(x)Φk(x)-Pk(j)(x) and ak(j)(x):=gk(j)(x)/λk, where λk:=b|Bk+1|αk+1/n∑l=k-1k+1‍(GNf)χlLp(·) and b is a constant which will be chosen later. Note that suppak(j)⊂Bk+1, I(j)=∑k=-∞∞‍λkak(j)(x).
Now we estimategk(j)Lp(·). To do this, let {ϕγk:|γ|⩽m} be the orthogonal polynomials restricted to A~k,ɛ with respect to the weight 1/|A~k,ɛ|, which are obtained from {xβ:|β|⩽m} by the Gram-Schmidt method, which means(25)ϕνk,ϕμk=1A~k,ɛ∫A~k,ɛ‍ϕνk(x)ϕμk(x)dx=δνμ,where δνμ=1 for ν=μ, otherwise 0.
It is easy to see thatPk(j)(x)=∑|γ|⩽m‍〈f(j)Φk,ϕγk〉ϕγk(x) for x∈A~k,ɛ. On the other hand, from (1/A~k,ɛ)∫A~k,ɛ‍ϕνk(x)ϕμk(x)dx=δνμ we infer that (26)1A~1,ɛ∫A~1,ɛ‍ϕνk(2k-1y)ϕμk(2k-1y)dy=δνμ.Thus, we deduce ϕνk(2k-1y)=ϕν1(y) a.e. That is, ϕνk(x)=ϕν1(21-kx) almost everywhere for x∈A~k,ɛ. Therefore |ϕνk(x)|⩽C for x∈A~k,ɛ. By the generalized Hölder inequality we have(27)Pkjx≲1A~k,ɛ∫A~k,ɛ‍fjxΦkxdx≲1A~k,ɛf(j)ΦkLp(·)χA~k,ɛLp′(·).By Lemma 5 we have (28)gk(j)Lp(·)⩽f(j)ΦkLp(·)+Pk(j)Lp(·)≲f(j)ΦkLp(·)+1|A~k,ɛ|f(j)ΦkLp(·)×χA~k,ɛLp′(·)χA~k,ɛLp(·)≲f(j)ΦkLp(·)+f(j)ΦkLp(·)≲(f∗ϕ(j))ΦkLp(·)⩽C∑l=k-1k+1‍(GNf)χlLp(·).Choose b=C; then ak(j)Lp(·)⩽|Bk+1|-αk+1/n and each ak(j) is a central (α(·),p(·))-atom with support contained in Bk+1. Here and below we abuse αk:=α2k and it is well defined in Definition 11. Thus,(29)sup⁡L∈Z2-Lλq∑k=-∞L‍λkq≲sup⁡L∈Z2-Lλq∑k=-∞L‍Bk+1qαk+1/n∑l=k-1k+1‍(GNf)χlLp(·)q≲sup⁡L∈Z2-Lλq∑k=-∞L‍Bk+1qαk+1/n(GNf)χkLp(·)q:=A.
Now we estimateA. By the condition of α(·) and Proposition 7 we consider it in two cases.
Case 1 (L⩽0). Consider (30)2-Lλq∑k=-∞L‍Bk+1qαk+1/n(GNf)χkLp(·)q≲sup⁡L⩽0,L∈Z2-Lλq∑k=-∞L‍2kqα(0)(GNf)χkLp(·)q≲GNfMKp(·),λα(·),qq.
Case 2 (L>0). Consider(31)2-Lλq∑k=-∞L‍Bk+1qαk+1/n(GNf)χkLp(·)q=2-Lλq∑k=-∞-2‍2(k+1)qα(0)(GNf)χkLp(·)q+2-Lλq∑k=-1L‍2(k+1)qα(∞)(GNf)χkLp(·)q≲2-Lλq∑k=-∞-1‍2kqα(0)(GNf)χkLp(·)q+2-Lλq∑k=0L‍2kqα(∞)(GNf)χkLp(·)q≲GNfMKp(·),λα(·),qq.Hence, (32)sup⁡L∈Z2-Lλq∑k=-∞L‍λkq≲GNfMKp(·),λα(·),qp.
It remains to estimateII(j). Let {ψdk:|γ|⩽m} be the dual basis of {xβ:|β|⩽m} with respect to the weight 1/|A~k,ɛ| on A~k,ɛ, that is, (33)ψγk,xβ=1A~k,ɛ∫A~k,ɛ‍xβψγk(x)dx=δβγ.Similar to the method of [48], let(34)hk,γjx:=∑l=-∞k‍ψγk(x)χA~k,ɛ(x)A~k,ɛ-ψγk+1(x)χA~k+1,ɛ(x)A~k+1,ɛ×∫Rn‍f(j)(y)Φl(y)yγdy.We write(35)II(j)=∑k=-∞∞‍∑γ⩽m‍fjΦk,xγψγk(x)χA~k,ɛ(x)=∑|γ|⩽m‍∑k=-∞∞‍∫Rn‍f(j)Φkxγdxψγk(x)χA~k,ɛ(x)A~k,ɛ=∑|γ|⩽m‍∑k=-∞∞‍∑l=-∞k‍∫Rn‍f(j)(x)Φl(x)xγdx×ψγk(x)χA~k,ɛ(x)A~k,ɛ-ψγk+1(x)χA~k+1,ɛ(x)A~k+1,ɛ=:∑|γ|⩽m‍∑k=-∞∞‍αk,γhk,γ(j)(x)αk,γ=:∑|γ|⩽m‍∑k=-∞∞‍αk,γak,γ(j)(x),where(36)αk,γ:=b~∑l=k-1k+1‍(GNf)χlLp(·)Bk+2αk+2/nand b~ is a constant which will be chosen later. Note that (37)∫Rn‍∑l=-∞k‍Φlxxγdx=∑l=-∞k‍∫A~k,ɛ‍Φlxxγdx≲2k(n+|γ|).By a computation we have (38)∫Rn‍f(j)(y)∑l=-∞k‍Φl(y)yγdy≲2kn+γGNfx,x∈Bk+2.Since (39)ψγk(x)χA~k,ɛ(x)|A~k,ɛ|-ψγk+1(x)χA~k+1,ɛ(x)|A~k+1,ɛ|≲2-k(n+|γ|)∑l=k-1k+1‍χl(x),it follows that (40)hk,γ(j)Lp(·)⩽C∑l=k-1k+1‍(GNf)χlLp(·).Take b~=C. It is easy to show that each ak,γ(j) is a central (α(·),p(·))-atom with support contained in A~k,ɛ∪A~k+1,ɛ⊂Bk+2, and(41)αk,γ=C∑l=k-1k+1‍(GNf)χlLp(·)Bk+2αk+2/n,where C is a constant independent of j, f, k, and γ. Moreover, we have(42)sup⁡L∈Z2-Lλq∑k=-∞L‍∑|γ|⩽m‍αk,γq≲sup⁡L∈Z2-Lλq∑k=-∞L‍Bk+2qαk+2/n∑l=k-1k+1‍(GNf)χlLp(·)q≲sup⁡L∈Z2-Lλq∑k=-∞L‍Bk+2qαk+2/n(GNf)χkLp(·)q:=B.Using the same argument as before for A, we obtain(43)B≲GNfMK˙p(·),λα(·),qq.Therefore, (44)sup⁡L∈Z2-Lλq∑k=-∞L‍∑|γ|⩽m‍αk,γq≲GNfMKp(·),λα(·),qp.Thus, we obtain that(45)f(j)(x)=∑d=-∞∞‍λdad(j)(x),where each ad(j) is a central (α(·),p(·))-atom with support contained in A~d,ɛ∪A~d+1,ɛ⊂Bd+2, λd is independent of j and(46)sup⁡L∈Z2-Lλ∑d=-∞L‍|λd|q1/q⩽CGNfMK˙p(·),λα(·),q<∞,where C is independent of j and f.
Since(47)sup⁡j∈N0a0(j)Lp(·)⩽B2-α2/n,by the Banach-Alaoglu theorem we obtain a subsequence a0jn0 of a0j converging in the weak ∗ topology of Lp(·) to some a0∈Lp(·). It is easy to verify that a0 is a central (α(·),p(·))-atom supported on B2. Next, since (48)sup⁡jn0∈N0a1(jn0)Lp(·)⩽B3-α3/n,another application of the Banach-Alaoglu theorem yields a subsequence a1jn1 of a1jn0 which converges weak ∗ in Lp(·) to a central (α(·),p(·))-atom a1 with support in B3. Furthermore, (49)sup⁡jn1∈N0a-1(jn1)Lp(·)⩽B1-α1/n.Similarly, there exists a subsequence a-1jn-1 of a-1jn1 which converges weak ∗ in Lp(·) to some a-1∈Lp(·), and a-1 is a central (α(·),p(·))-atom supported on B1. Repeating the above procedure for each d∈Z, we can find a subsequence adjnd of adj converging weak ∗ in Lp(·) to some ad∈Lp(·) which is a central (α(·),p(·))-atom supported on Bd+2. By using the diagonal method we obtain a subsequence {jν} of N0 such that, for each d∈Z, limν→∞ad(jν)=ad in the weak ∗ topology of Lp(·) and therefore in S′(Rn).
Now we only need to prove thatf=∑d=-∞∞‍λdad in the sense of S′(Rn). For each φ∈S(Rn), note that suppad(jν)⊂(A~d,ɛ∪A~d+1,ɛ)⊂(Ad-1∪Ad∪Ad+1∪Ad+2). Using the same argument in [48], we have (50)f,φ=limν→∞∑d=-∞∞‍λd∫Rn‍ad(jν)(x)φ(x)dx.Recall that m=max⁡{[α(0)-nδ2],[α∞-nδ2]}. If d⩽0, then by Lemmas 3 and 4 we have(51)∫Rn‍ad(jν)(x)φ(x)dx=∫Rn‍ad(jν)(x)φ(x)-∑|β|⩽m‍Dβφ(0)β!xβdx≲∫Rn‍adjνx·xm+1dx≲2dm+1∫Rn‍adjνxdx≲2d(m+1-αd+2)χBd+2Lp′(·)≲2d(m+1-αd+2)Bd+2B2δ2χB2Lp′(·)≲2d(m+1-αd+2+nδ2)B2B0χB0Lp′(·)≲2d(m+1-αd+2+nδ2).
Ifd>0, let k0∈N0 such that min⁡{k0+α(0)-n,k0+α∞-n}>0; then by Lemmas 4 and 3 again we have (52)∫Rn‍ad(jν)(x)φ(x)dx≲∫Rn‍adjνxx-k0dx≲2-d(k0+αd+2)χBd+2Lp′(·)≲2-d(k0+αd+2)Bd+2B0χB0Lp′(·)≲2-d(k0+αd+2-n).
Let(53)μd=λd2d(m+1-αd+2+nδ2),d⩽0λd2-d(k0+αd+2-n),d>0.Then(54)sup⁡L∈Z2-Lλ∑d=-∞L‍μd≲sup⁡L∈Z2-Lλq∑d=-∞L‍λdq1/q≲GNfMK˙p·,λα·,q<∞,λd∫Rn‍adjνxφxdx≲μd,which implies that(55)f,φ=∑d=-∞∞‍limν→∞λd∫Rn‍ad(jν)(x)φ(x)dx=∑d=-∞∞‍λd∫Rn‍ad(x)φ(x)dx.This establishes the identity we wanted.
To prove the sufficiency, for convenience, we denotesup⁡L∈Z2-Lλ∑k=-∞L‍|λk|q=Λ. Firstly we have(56)GNfMK˙p(·),λα(·),qq≈max⁡sup⁡L⩽0,L∈Z2-Lλq∑k=-∞L‍2kqα(0)(GNf)χkLp(·)q,sup⁡L>0,L∈Z2-Lλq∑k=-∞-1‍2kqα(0)(GNf)χkLp(·)q+∑k=0L‍2kqα(∞)(GNf)χkLp(·)q≲max⁡I,II+III,where (57)I:=sup⁡L⩽0,L∈Z2-Lλq∑k=-∞L‍2kqα(0)(GNf)χkLp(·)qII:=∑k=-∞-1‍2kqα(0)(GNf)χkLp(·)qIII:=sup⁡L>0,L∈Z2-Lλq∑k=0L‍2kqα(∞)(GNf)χkLp(·)q.Now we have(58)I=sup⁡L⩽0,L∈Z2-Lλq∑k=-∞L‍2kqα(0)(GNf)χkLp(·)q≲sup⁡L⩽0,L∈Z2-Lλq∑k=-∞L‍2kqα0∑l=k∞‍λlalLp·q+sup⁡L⩽0,L∈Z2-Lλq∑k=-∞L‍2kqα(0)∑l=-∞k-1‍λl(GNal)χkLp(·)q:=I1+I2.II=∑k=-∞-1‍2kqα(0)(GNf)χkLp(·)q≲∑k=-∞-1‍2kqα(0)∑l=k∞‍λlalLp(·)q+∑k=-∞-1‍2kqα(0)∑l=-∞k-1‍λl(GNal)χkLp(·)q:=II1+II2III=sup⁡L>0,L∈Z2-Lλq∑k=0L‍2kqα∞(GNf)χkLp(·)q≲sup⁡L>0,L∈Z2-Lλq∑k=0L‍2kqα∞∑l=kL‍λlalLp(·)q+sup⁡L>0,L∈Z2-Lλq∑k=0L‍2kqα∞∑l=-∞k-1‍λl(GNal)χkLp(·)q:=III1+III2.
To estimateI, II, and III we need a pointwise estimate for GNal(x) on Dk, where k⩾l+2. Let ϕ∈AN, m∈N such that αk-nδ2<m+1. Denote by Pm the mth order Taylor series expansion of ϕ at y/t. If |x-y|<t, then from the vanishing moment condition of al we have(59)al∗ϕty=t-n∫Rn‍al(z)ϕy-zt-Pm-ztdz≲t-n∫Rn‍alzztm+11+(y-θz)t-(n+m+1)dz≲∫Rn‍alzzm+1t+y-θz-(n+m+1)dz,where 0<θ<1. Since x∈Dk for k∈Z, we have |x|⩾2k-1. From |x-y|<t and |z|<2l, we have (60)t+y-θz⩾x-y+y-θz⩾x-z⩾x2.Thus,(61)al∗ϕty≲∫Rn‍alzzm+1x-y+y-θz-(n+m+1)dz≲2lm+1x-n+m+1∫Rn‍azdz≲2l(m+1)2-k(n+m+1)Bl-αl/nχBlLp′(·).Therefore, we have(62)GNal(x)≲2lm+12-kn+m+1Bl-αl/nχBlLp′·,x∈Dk,k⩾l+2.
To proceed, we consider them into two cases0<q⩽1 and 1<q<∞.
If0<q⩽1, (63)I1=sup⁡L⩽0,L∈Z2-Lλq∑k=-∞L‍2α(0)kq∑l=k∞‍λl(Tbl)χkLp(·)q≲sup⁡L⩽0,L∈Z2-Lλq∑k=-∞L‍2α(0)kq∑l=k∞‍λl2-αllq≲sup⁡L⩽0,L∈Z2-Lλq∑k=-∞L‍2α0kq×∑l=k-1‍λlq2-α(0)lq+∑l=0∞‍λlq2-α∞lq≲sup⁡L⩽0,L∈Z2-Lλq∑k=-∞L‍∑l=k-1‍λlq2α(0)(k-l)q+sup⁡L⩽0,L∈Z2-Lλq∑k=-∞L‍2α(0)kq∑l=0∞‍λlq2-α∞lq≲sup⁡L⩽0,L∈Z2-Lλq∑l=-∞-1‍λlq∑k=-∞l‍2α(0)(k-l)q+sup⁡L⩽0,L∈Z∑l=0∞‍2-lλqλlq2(λ-α∞)lq2-Lλq∑k=-∞L‍2α(0)kq≲Λ+sup⁡L⩽0,L∈Z2-Lλq∑l=L-1‍λlq∑k=-∞l‍2α(0)(k-l)q+Λsup⁡L⩽0,L∈Z∑l=0∞‍2(λ-α∞)lq∑k=-∞L‍2α(0)kq-Lλq≲Λ+sup⁡L⩽0,L∈Z∑l=L-1‍2-lλqλlq2(l-L)λq∑k=-∞l‍2α(0)(k-l)q+Λ≲Λ+Λsup⁡L⩽0,L∈Z∑l=L-1‍2(l-L)λq∑k=-∞l‍2α(0)(k-l)q≲Λ.I2=sup⁡L⩽0,L∈Z2-Lλq∑k=-∞L‍2kqα(0)∑l=-∞k-1‍λl(GNal)χkLp(·)q≲sup⁡L⩽0,L∈Z2-Lλq×∑k=-∞L‍2kqα(0)∑l=-∞k-1‍λl2l(m+1)-k(n+m+1)∑l=-∞k-1‍×Bl-αl/nχBlLp′(·)χBkLp(·)q≲sup⁡L⩽0,L∈Z2-Lλq∑k=-∞L‍∑l=-∞k-1‍λl2(l-k)(m+1+nδ2)-(l-k)α(0)q≲sup⁡L⩽0,L∈Z2-Lλq∑k=-∞L‍∑l=-∞k-1‍λl2(l-k)(m+1+nδ2-α(0))q=sup⁡L⩽0,L∈Z2-Lλq∑l=-∞L-1‍∑k=l+1L‍λlq2(l-k)(m+1+nδ2-α(0))q≲sup⁡L⩽0,L∈Z2-Lλq∑l=-∞L‍λlq≲Λ.Then we turn to estimate II: (64)II1=∑k=-∞-1‍2kqα(0)∑l=k∞‍λlalLp(·)q≲∑k=-∞-1‍2kqα(0)∑l=k∞‍λlBl-αl/nq≲∑k=-∞-1‍2α(0)kq∑l=k-1‍λlq2-α(0)lq+∑l=0∞‍λlq2-α∞lq≲∑k=-∞-1‍∑l=k-1‍λlq2α0k-lq+∑k=-∞-1‍2α(0)kq∑l=0∞‍λlq2-α∞lq≲∑l=-∞-1‍λlq∑k=-∞l‍2α(0)(k-l)q+∑l=0∞‍λlq2-α∞lq∑k=-∞-1‍2α(0)kq≲∑l=-∞-1‍λlq+∑l=0∞‍2-lλqλlq2-α∞lq∑k=-∞-1‍2α(0)kq≲Λ+Λ∑i=-∞l‍λiq∑l=0∞‍2(λ-α∞)lq∑k=-∞l‍2α(0)kq≲Λ.(65)II2=∑k=-∞-1‍2kqα(0)∑l=-∞k-1‍λl(GNal)χkLp(·)q≲∑k=-∞-1‍2kqα(0)∑l=-∞k-1‍λl2l(m+1)-k(n+m+1)∑l=-∞k-1‍×Bl-αl/nχBlLp′(·)χBkLp(·)q≲∑k=-∞-1‍∑l=-∞k-1‍λl2(l-k)(m+1+nδ2)-(l-k)α(0)q≲∑k=-∞-1‍∑l=-∞k-1‍λl2(l-k)(m+1+nδ2-α(0))q≲∑k=-∞-1‍∑l=-∞k-1‍λlq2(l-k)(m+1+nδ2-α(0))q=∑l=-∞-2‍∑k=l+1-1‍λlq2(l-k)(m+1+nδ2-α(0))q≲∑l=-∞-1‍λlq⩽Λ.Third, we estimate III:(66)III1=sup⁡L>0,L∈Z2-Lλq∑k=0L‍2kqα∞Bkα∞q/n∑l=k∞‍λlalLp(·)q≲sup⁡L>0,L∈Z2-Lλq∑k=0L‍2kqα∞Bkα∞q/n∑l=k∞‍λlBl-αl/nq≲sup⁡L>0,L∈Z2-Lλq∑k=0L‍∑l=k∞‍Bkα∞q/nλlqBl-α∞q/n≲sup⁡L>0,L∈Z2-Lλq∑k=0L‍∑l=k∞‍λlq2(k-l)α∞q=sup⁡L>0,L∈Z2-Lλq×∑l=0L‍λlq∑k=0l‍2(k-l)α∞q+∑l=L∞‍λlq∑k=0L‍2(k-l)α∞q≲sup⁡L>0,L∈Z2-Lλq∑l=0L‍λlq+sup⁡L>0,L∈Z∑l=L∞‍2(lλq-Lλq)2-lλq∑i=-∞l‍λiq∑k=0L‍2(k-l)α∞q/2≲Λ+Λsup⁡L>0,L∈Z∑l=L∞‍2(l-L)λq2(L-l)α∞q/2≲Λ+Λsup⁡L>0,L∈Z∑l=L∞‍2(l-L)q(λ-α∞/2)≲Λ.III2=sup⁡L>0,L∈Z2-Lλq∑k=0L‍2kqα∞∑l=-∞k-1‍λl(GNal)χkLp(·)q≲sup⁡L>0,L∈Z2-Lλq×∑k=0L‍2kqα∞∑l=-∞k-1‍λl2l(m+1)-k(n+m+1)∑l=-∞k-1‍×Bl-αl/nχBlLp′(·)χBkLp(·)q≲sup⁡L>0,L∈Z2-Lλq∑k=0L‍∑l=-∞k-1‍λl2(l-k)(m+1+nδ2)-lαl+kα∞q≲sup⁡L>0,L∈Z2-Lλq∑k=0L‍∑l=-∞-1‍λl2(l-k)(m+1+nδ2)-lα(0)+kα∞q+sup⁡L>0,L∈Z2-Lλq∑k=0L‍∑l=0k-1‍λl2(l-k)(m+1+nδ2-α∞)q≲sup⁡L>0,L∈Z2-Lλq∑k=0L‍2qk[α∞-(m+1+nδ2)]×∑l=-∞-1‍λl2l(m+1+nδ2-α(0))q+sup⁡L>0,L∈Z2-Lλq∑k=0L‍∑l=0k-1‍λlq2(l-k)(m+1+nδ2-α∞)q≲sup⁡L>0,L∈Z2-Lλq∑l=-∞-1‍λlq2l(m+1+nδ2-α(0))q+sup⁡L>0,L∈Z2-Lλq∑l=0L-1‍∑k=l+1L‍λlq2(l-k)(m+1+nδ2-α∞)q≲sup⁡L>0,L∈Z2-Lλq∑l=-∞-1‍λlq+sup⁡L>0,L∈Z2-Lλq∑l=0L‍λlq=sup⁡L>0,L∈Z2-Lλq∑l=-∞L‍λlq.If 1<q<∞, we have(67)I1=sup⁡L⩽0,L∈Z2-Lλq∑k=-∞L‍2kqα(0)∑l=k∞‍λlalLp(·)q≲sup⁡L⩽0,L∈Z2-Lλq∑k=-∞L‍2kqα(0)∑l=k∞‍λlBl-αl/nq≲sup⁡L⩽0,L∈Z2-Lλq∑k=-∞L‍∑l=k-1‍λl2α(0)(k-l)q+sup⁡L⩽0,L∈Z2-Lλq∑k=-∞L‍2α(0)kq∑l=0∞‍λl2-α∞lq≲sup⁡L⩽0,L∈Z2-Lλq∑k=-∞L‍∑l=k-1‍λlq2α0k-lq/2×∑l=k-1‍2α0k-lq′/2q/q′+sup⁡L⩽0,L∈Z2-Lλq∑k=-∞L‍2α(0)kq∑l=0∞‍λlq2-α∞jq/2×∑l=0∞‍2-α∞jq′/2q/q′≲sup⁡L⩽0,L∈Z2-Lλq∑k=-∞L‍∑l=k-1‍λlq2α(0)(k-l)q/2+sup⁡L⩽0,L∈Z2-Lλq∑k=-∞L‍2α(0)kq∑l=0∞‍λlq2-α∞lq/2≲sup⁡L⩽0,L∈Z2-Lλq∑l=-∞-1‍λlq∑k=-∞l‍2α(0)(k-l)q/2+sup⁡L⩽0,L∈Z∑l=0∞‍2-lλqλlq2(λ-α∞/2)lq2-Lλq∑k=-∞L‍2α(0)kq⩽sup⁡L⩽0,L∈Z2-Lλq∑l=-∞L‍λlq+sup⁡L⩽0,L∈Z2-Lλq∑l=L-1‍λlq∑k=-∞l‍2α(0)(k-l)q/2+Λsup⁡L⩽0,L∈Z∑l=0∞‍2(λ-α∞/2)lq∑k=-∞L‍2α(0)kq-Lλq≲Λ+sup⁡L⩽0,L∈Z∑l=L-1‍2-lλqλlq2(l-L)λq×∑k=-∞l‍2α(0)(k-l)q/2+Λ≲Λ+Λsup⁡L⩽0,L∈Z∑l=L-1‍2l-Lλq∑k=-∞j‍2α0k-lq/2≲Λ.I2=sup⁡L⩽0,L∈Z2-Lλq∑k=-∞L‍2kqα(0)∑l=-∞k-1‍λl(GNal)χkLp(·)q≲sup⁡L⩽0,L∈Z2-Lλq×∑k=-∞L‍2kqα(0)∑l=-∞k-1‍λl2l(m+1)-k(n+m+1)∑l=-∞k-1‍×Bl-αl/nχBlLp′(·)χBkLp(·)q≲sup⁡L⩽0,L∈Z2-Lλq∑k=-∞L‍∑l=-∞k-1‍λl2(l-k)(m+1+nδ2)-(l-k)α(0)q≲sup⁡L⩽0,L∈Z2-Lλq∑k=-∞L‍∑l=-∞k-1‍λl2(l-k)(m+1+nδ2-α(0))q≲sup⁡L>0,L∈Z2-Lλq∑k=-∞L‍∑l=-∞k-1‍λlq2(l-k)(m+1+nδ2-α(0))q/2×∑l=-∞k-1‍2(l-k)(m+1+nδ2-α(0))q′/2q/q′≲sup⁡L⩽0,L∈Z2-Lλq∑k=-∞L‍∑l=-∞k-1‍λlq2(l-k)(m+1+nδ2-α(0))q/2=sup⁡L⩽0,L∈Z2-Lλq∑l=-∞L-1‍∑k=l+1L‍λlq2(l-k)(m+1+nδ2-α(0))q/2≲sup⁡L⩽0,L∈Z2-Lλq∑l=-∞L‍λlq⩽Λ.Second, we estimate II. As the same argument before, we obtain that (68)II1=∑k=-∞-1‍2kqα(0)∑l=k∞‍λlalLp(·)q≲∑k=-∞-1‍2kqα(0)∑l=k∞‍λlBl-αl/nq≲∑k=-∞-1‍∑j=k-1‍λj2α(0)(k-j)q+∑k=-∞-1‍2α(0)kq∑j=0∞‍λj2-α∞jq≲∑k=-∞-1‍∑l=k-1‍λlq2α(0)(k-l)q/2∑l=k-1‍2α(0)(k-l)q′/2q/q′+∑k=-∞-1‍2α(0)kq∑l=0∞‍λjq2-α∞lq/2∑l=0∞‍2-α∞lq′/2q/q′≲∑k=-∞-1‍λlq∑k=-∞l‍2α(0)(k-l)q/2+∑l=0∞‍λlq2-α∞lq/2∑k=-∞-1‍2α(0)kq≲∑l=-∞-1‍λlq+∑l=0∞‍2(λ-α∞/2)lq2-lλq∑i=-∞l‍λiq∑k=-∞-1‍2α(0)kq⩽Λ+Λ∑l=0∞‍2(λ-α∞/2)lq∑k=-∞-1‍2α(0)kq≲Λ.II2=∑k=-∞-1‍2kqα(0)∑l=-∞k-1‍λl(GNal)χkLp(·)q≲∑k=-∞-1‍2kqα(0)∑l=-∞k-1‍λl2l(m+1)-k(n+m+1)∑l=-∞k-1‍×Bl-αl/nχBlLp′(·)χBkLp(·)q≲∑k=-∞-1‍∑l=-∞k-1‍λl2(l-k)(m+1+nδ2)-(l-k)α(0)q≲∑k=-∞-1‍∑l=-∞k-1‍λl2(l-k)(m+1+nδ2-α(0))q≲∑k=-∞-1‍∑l=-∞k-1‍λlq2(l-k)(m+1+nδ2-α(0))q/2×∑l=-∞k-1‍2(l-k)(m+1+nδ2-α(0))q′/2q/q′≲∑k=-∞-1‍∑l=-∞k-1‍λlq2(l-k)(m+1+nδ2-α(0))q/2=∑l=-∞-2‍∑k=l+1-1‍λlq2(l-k)(m+1+nδ2-α(0))q/2≲∑l=-∞-1‍λlq≲Λ.Third, we estimate III. We have (69)III1=sup⁡L>0,L∈Z2-Lλq∑k=0L‍2kqα∞Bkα∞q/n∑l=k∞‍λlalLp(·)q≲sup⁡L>0,L∈Z2-Lλq∑k=0L‍2kqα∞Bkα∞q/n∑l=k∞‍λlBl-αl/nq≲sup⁡L>0,L∈Z2-Lλq∑k=0L‍2kqα∞Bkα∞q/n∑l=k∞‍λlqBl-αlq/(2n)×∑l=k∞‍Bl-αlq′/(2n)q/q′≲sup⁡L>0,L∈Z2-Lλq∑k=0L‍∑l=k∞‍Bkα∞q/(2n)λlqBl-α∞q/(2n)≲sup⁡L>0,L∈Z2-Lλq∑k=0L‍∑l=k∞‍λlq2(k-l)α∞q/2=sup⁡L>0,L∈Z2-Lλq∑l=0L‍λlq∑k=0l‍2k-lα∞q/2+∑l=L∞‍λlq∑k=0L‍2k-lα∞q/2≲sup⁡L>0,L∈Z2-Lλq∑l=0L‍λlq+sup⁡L>0,L∈Z∑l=L∞‍2(jλq-Lλq)2-lλq∑i=-∞l‍λiq∑k=0L‍2(k-l)α∞q/2≲Λ+Λsup⁡L>0,L∈Z∑l=L∞‍2(l-L)λq2(L-l)α∞q/2≲Λ+Λsup⁡L>0,L∈Z∑l=L∞‍2(l-L)q(λ-α∞/2)≲Λ.III2=sup⁡L>0,L∈Z2-Lλq∑k=0L‍2kqα∞∑l=-∞k-1‍λl(GNal)χkLp(·)q≲sup⁡L>0,L∈Z2-Lλq×∑k=0L‍2kqα∞∑l=-∞k-1‍λl2l(m+1)-k(n+m+1)∑l=-∞k-1‍×Bl-αl/nχBlLp′(·)χBkLp(·)q≲sup⁡L>0,L∈Z2-Lλq∑k=0L‍∑l=-∞k-1‍λl2(l-k)(m+1+nδ2)-lαl+kα∞q≲sup⁡L>0,L∈Z2-Lλq∑k=0L‍∑l=-∞-1‍λl2(l-k)(m+1+nδ2)-lα(0)+kα∞q+sup⁡L>0,L∈Z2-Lλq∑k=0L‍∑l=0k-1‍λl2(l-k)(m+1+nδ2-α∞)q≲sup⁡L>0,L∈Z2-Lλq∑k=0L‍2qkα∞-m+1+nδ2×∑l=-∞-1‍λl2l(m+1+nδ2-α(0))q+sup⁡L>0,L∈Z2-Lλq∑k=0L‍∑l=0k-1‍λlq2(l-k)(m+1+nδ2-α∞)q/2×∑l=0k-1‍2(l-k)(m+1+nδ2-α∞)q′/2q/q′≲sup⁡L>0,L∈Z2-Lλq∑l=-∞-1‍λlq2l(m+1+nδ2-α(0))q/2×∑l=-∞-1‍2l(m+1+nδ2-α(0))q′/2q/q′+sup⁡L>0,L∈Z2-Lλq∑k=0L‍∑l=0k-1‍λlq2(l-k)(m+1+nδ2-α∞)q/2≲sup⁡L>0,L∈Z2-Lλq∑l=-∞-1‍λlq2l(m+1+nδ2-α(0))q/2+sup⁡L>0,L∈Z2-Lλq∑l=0L-1‍∑k=l+1L‍λlq2(l-k)(m+1+nδ2-α∞)q/2≲sup⁡L>0,L∈Z2-Lλq∑l=-∞-1‍λlq+sup⁡L>0,L∈Z2-Lλq∑l=0L‍λlq=sup⁡L>0,L∈Z2-Lλq∑l=-∞L‍λlq⩽Λ.Thus, we finish the proof of Theorem 13.


## 3. Applications

As an application of the atomic decompositions, we will prove the following result.Theorem 14.
Let0<q<∞, 0≤λ<∞, p(·)∈B(Rn), and α(·)∈L∞(Rn) be log-Hölder continuous both at the origin and infinity, 2λ≤α(·), nδ2⩽α(0), α∞<∞, nonnegative integer and s=max⁡{[α(0)-nδ2],[α∞-nδ2]}, and δ2 as in Lemma 4. If a sublinear operator T satisfies that(i)
T is bounded on Lp(·);(ii)
there exists a constantδ>0 such that s+δ>max⁡{α(0)-nδ2,α∞-nδ2}, and for any compact support function f with (70)∫Rn‍fxxβdx=0,β⩽s,Tf satisfies the size condition(71)Tfx⩽Cdiam⁡suppfs+δx-n+s+δf1,whendist⁡x,suppf⩾x2.Then there exists a constant C such that(72)TfMK˙p(·),λα(·),q⩽CfHMK˙p·,λα·,q,TfMKp(·),λα(·),q⩽CfHMKp(·),λα(·),qfor f∈HMK˙p(·),λα(·),q and f∈HMKp(·),λα(·),q, respectively.Proof.
It suffices to prove the homogeneous case. Supposef∈HMK˙p(·),λα(·),q. By Theorem 13, f=∑j=-∞∞‍λjbj converges in S′(Rn), where each bj is a central (α(·),q(·))-atom with support contained in Bj and(73)fHMK˙p(·),λα(·),q≈inf⁡sup⁡L∈Z2-Lλ∑j=-∞L‍λjq1/q.For simplicity, we denote Λ=sup⁡L∈Z2-Lλ∑j=-∞L‍|λj|q. By Proposition 7, we have (74)TfMK˙p(·)λα(·),qq≈max⁡sup⁡L⩽0,L∈Z2-Lλq∑k=-∞L‍2kqα(0)(Tf)χkLp(·)q,sup⁡L>0,L∈Z2-Lλq∑k=-∞-1‍2kqα(0)(Tf)χkLp(·)q+∑k=0L‍2kqα(∞)(Tf)χkLp(·)q≲max⁡I,II+III,where (75)I:=sup⁡L⩽0,L∈Z2-Lλq∑k=-∞L‍2kqα(0)(Tf)χkLp(·)q,II:=∑k=-∞-1‍2kqα(0)(Tf)χkLp(·)q,III:=sup⁡L>0,L∈Z2-Lλq∑k=0L‍2kqα(∞)(Tf)χkLp(·)q.
To complete our proof, we only need show that there exists a positive constantC such that I,II,III⩽CΛ.
First, we estimateI:(76)I=sup⁡L⩽0,L∈Z2-Lλq∑k=-∞L‍2kqα(0)(Tf)χkLp(·)q≲sup⁡L⩽0,L∈Z2-Lλq∑k=-∞L‍2kqα(0)∑j=k∞‍λj(Tbj)χkLp(·)q+sup⁡L⩽0,L∈Z2-Lλq∑k=-∞L‍2kqα(0)∑j=-∞k-1‍λj(Tbj)χkLp(·)q:=I1+I2.By the boundedness of T in Lp(·), we have(77)(Tbj)χkLp(·)⩽bjLp(·)⩽Bj-αj/n=2-αjj.Therefore, when 0<q⩽1, we get (78)I1=sup⁡L⩽0,L∈Z2-Lλq∑k=-∞L‍2α(0)kq∑j=k∞‍λj(Tbj)χkLp(·)q≲sup⁡L⩽0,L∈Z2-Lλq∑k=-∞L‍2α(0)kq∑j=k∞‍λj2-αjjq≲sup⁡L⩽0,L∈Z2-Lλq×∑k=-∞L‍2α(0)kq∑j=k-1‍λjq2-α(0)jq+∑j=0∞‍λjq2-α∞jq≲sup⁡L⩽0,L∈Z2-Lλq∑k=-∞L‍∑j=k-1‍λjq2α(0)(k-j)q+sup⁡L⩽0,L∈Z2-Lλq∑k=-∞L‍2α(0)kq∑j=0∞‍λjq2-α∞jq≲sup⁡L⩽0,L∈Z2-Lλq∑j=-∞-1‍λjq∑k=-∞j‍2α(0)(k-j)q+sup⁡L⩽0,L∈Z∑j=0∞‍2-jλqλjq2(λ-α∞)jq2-Lλq∑k=-∞L‍2α(0)kq⩽sup⁡L⩽0,L∈Z2-Lλq∑j=-∞L‍λjq+sup⁡L⩽0,L∈Z2-Lλq∑j=L-1‍λjq∑k=-∞j‍2α(0)(k-j)q+Λsup⁡L⩽0,L∈Z∑j=0∞‍2(λ-α∞)jq∑k=-∞L‍2α(0)kq-Lλq≲Λ+sup⁡L⩽0,L∈Z∑j=L-1‍2-jλqλjq2(j-L)λq∑k=-∞j‍2α(0)(k-j)q+Λ≲Λ+Λsup⁡L⩽0,L∈Z∑j=L-1‍2(j-L)λq∑k=-∞j‍2α(0)(k-j)q≲Λ.When 1<q<∞, let 1/q+1/q′=1 and we obtain (79)I1=sup⁡L⩽0,L∈Z2-Lλq∑k=-∞L‍2α(0)kq∑j=k∞‍λj(Tbj)χkLp(·)q≲sup⁡L⩽0,L∈Z2-Lλq∑k=-∞L‍2α(0)kq∑j=k∞‍|λj|2-αjjq≲sup⁡L⩽0,L∈Z2-Lλq∑k=-∞L‍∑j=k-1‍λj2α(0)(k-j)q+sup⁡L⩽0,L∈Z2-Lλq∑k=-∞L‍2α(0)kq∑j=0∞‍λj2-α∞jq≲sup⁡L⩽0,L∈Z2-Lλq∑k=-∞L‍∑j=k-1‍λjq2α0k-jq/2×∑j=k-1‍2α(0)(k-j)q′/2q/q′+sup⁡L⩽0,L∈Z2-Lλq∑k=-∞L‍2α0kq∑j=0∞‍λjq2-α∞jq/2×∑j=0∞‍2-α∞jq′/2q/q′≲sup⁡L⩽0,L∈Z2-Lλq∑k=-∞L‍∑j=k-1‍λjq2α(0)(k-j)q/2+sup⁡L⩽0,L∈Z2-Lλq∑k=-∞L‍2α(0)kq∑j=0∞‍λjq2-α∞jq/2≲sup⁡L⩽0,L∈Z2-Lλq∑j=-∞-1‍λjq∑k=-∞j‍2α(0)(k-j)q/2+sup⁡L⩽0,L∈Z∑j=0∞‍2-jλqλjq2(λ-α∞/2)jq2-Lλq∑k=-∞L‍2α(0)kq⩽sup⁡L⩽0,L∈Z2-Lλq∑j=-∞L‍λjq+sup⁡L⩽0,L∈Z2-Lλq∑j=L-1‍λjq∑k=-∞j‍2α(0)(k-j)q/2+Λsup⁡L⩽0,L∈Z∑j=0∞‍2(λ-α∞/2)jq∑k=-∞L‍2α(0)kq-Lλq≲Λ+sup⁡L⩽0,L∈Z∑j=L-1‍2-jλqλjq2(j-L)λq∑k=-∞j‍2α(0)(k-j)q/2+Λ≲Λ+Λsup⁡L⩽0,L∈Z∑j=L-1‍2(j-L)λq∑k=-∞j‍2α(0)(k-j)q/2≲Λ.So, we have I1≲Λ.
Second, we estimateI2. By (71) and Lemma 3, we get(80)Tbjx≲x-n+s+δ2js+δ∫Bj‍bjydy≲2-k(n+s+δ)2j(s+δ)bjLp(·)χBjLp′(·)≲2j(s+δ-αj)-k(s+δ+n)χBjLp′(·).So by Lemmas 3 and 4, we have(81)(Tbj)χkLp(·)≲2j(s+δ-αj)-k(s+δ+n)χBjLp′(·)χBkLp(·)≲2j(s+δ-αj)-k(s+δ)2-knBkχBkLp′(·)-1χBjLp′(·)≲2j(s+δ-αj)-k(s+δ)χBjLp′(·)χBkLp′(·)≲2(s+δ+nδ2)(j-k)-jαj.Therefore, when 0<q⩽1, by nδ2⩽α(0)<s+δ+nδ2 we get (82)I2=sup⁡L⩽0,L∈Z2-Lλq∑k=-∞L‍2α(0)kq∑j=-∞k-1‍λj(Tbj)χkLp(·)q≲sup⁡L⩽0,L∈Z2-Lλq×∑k=-∞L‍2α(0)kq∑j=-∞k-1‍λjq2[(s+δ+nδ2)(j-k)-jα(0)]q=sup⁡L⩽0,L∈Z2-Lλq∑j=-∞L‍λjq∑k=j+1-1‍2(j-k)(s+δ+nδ2-α(0))q≲Λ.When 1<q<∞, let 1/q+1/q′=1. Since nδ2⩽α(0)<s+δ+nδ2, by Hölder’s inequality, we have (83)I2=sup⁡L⩽0,L∈Z2-Lλq∑k=-∞L‍2α(0)kq∑j=-∞k-1‍λj(Tbj)χkLp(·)q≲sup⁡L⩽0,L∈Z2-Lλq∑k=-∞L‍2α(0)kq∑j=-∞k-1‍|λj|2(s+δ+nδ2)(j-k)-jα(0)q≲sup⁡L⩽0,L∈Z2-Lλq×∑k=-∞L‍2α(0)kq∑j=-∞k-1‍λjq2(j-k)(s+δ+nδ2-α(0))q/2×∑j=-∞k-1‍2(j-k)(s+δ+nδ2-α(0))q′/2q/q′≲sup⁡L⩽0,L∈Z2-Lλq×∑j=-∞L‍2α(0)kq∑j=-∞k-1‍λjq2(j-k)(s+δ+nδ2-α(0))q/2=sup⁡L⩽0,L∈Z2-Lλq∑j=-∞L‍λjq∑k=j+1-1‍2(j-k)(s+δ+nδ2-α(0))q/2≲Λ.Hence, we have I≲Λ.
Third, we estimateII. Consider(84)II=∑k=-∞-1‍2kqα(0)(Tf)χkLp(·)q≲∑k=-∞-1‍2kqα(0)∑j=k∞‍λj(Tbj)χkLp(·)q+∑k=-∞-1‍2kqα(0)∑j=-∞k-1‍λj(Tbj)χkLp(·)q:=II1+II2.
When0<q⩽1, we get(85)II1=∑k=-∞-1‍2α(0)kq∑j=k∞‍λj(Tbj)χkLp(·)q≲∑k=-∞-1‍2α(0)kq∑j=k∞‍λj2-αjjq≲∑k=-∞-1‍2α(0)kq∑j=k-1‍λjq2-α(0)jq+∑j=0∞‍λjq2-α∞jq≲∑k=-∞-1‍∑j=k-1‍λjq2α(0)(k-j)q+∑k=-∞-1‍2α(0)kq∑j=0∞‍λjq2-α∞jq≲∑j=-∞-1‍λjq∑k=-∞j‍2α(0)(k-j)q+∑j=0∞‍λjq2-α∞jq∑k=-∞-1‍2α(0)kq≲∑j=-∞-1‍λjq+∑j=0∞‍2-jλqλjq2-α∞jq∑k=-∞-1‍2α(0)kq≲Λ+Λ∑i=-∞j‍λiq∑j=0∞‍2(λ-α∞)jq∑k=-∞j‍2α(0)kq≲Λ.When 1<q<∞, let 1/q+1/q′=1 and we obtain(86)II1=∑k=-∞-1‍2α(0)kq∑j=k∞‍λj(Tbj)χkLp(·)q≲∑k=-∞-1‍2α(0)kq∑j=k∞‍λj2-αjjq≲∑k=-∞-1‍∑j=k-1‍λj2α(0)(k-j)q+∑k=-∞-1‍2α0kq∑j=0∞‍λj2-α∞jq≲∑k=-∞-1‍∑j=k-1‍λjq2α(0)(k-j)q/2∑j=k-1‍2α(0)(k-j)q′/2q/q′+∑k=-∞-1‍2α(0)kq∑j=0∞‍λjq2-α∞jq/2∑j=0∞‍2-α∞jq′/2q/q′≲∑k=-∞-1‍λjq∑k=-∞j‍2α(0)(k-j)q/2+∑j=0∞‍λjq2-α∞jq/2∑k=-∞-1‍2α(0)kq≲∑j=-∞-1‍λjq+∑j=0∞‍2(λ-α∞/2)jq2-jλq∑i=-∞j‍λiq∑k=-∞-1‍2α(0)kq⩽Λ+Λ∑j=0∞‍2(λ-α∞/2)jq∑k=-∞-1‍2α(0)kq≲Λ.For II2, when 0<q⩽1, by nδ2⩽α(0)<s+δ+nδ2 we get (87)II2=∑k=-∞-1‍2α0kq∑j=-∞k-1‍λjTbjχkLp·q≲∑k=-∞-1‍2α0kq∑j=-∞k-1‍λjq2s+δ+nδ2j-k-jα0q=∑k=-∞-1‍λjq∑k=j+1-1‍2j-ks+δ+nδ2-α0q≲∑k=-∞-1‍λjq⩽Λ.When 1<q<∞, let 1/q+1/q′=1. Since nδ2⩽α(0)<s+δ+nδ2, by Hölder’s inequality, we have (88)II2=∑k=-∞-1‍2α(0)kq∑j=-∞k-1‍λj(Tbj)χkLp(·)q≲∑k=-∞-1‍2α(0)kq∑j=-∞k-1‍λj2(s+δ+nδ2)(j-k)-jα(0)q≲∑k=-∞-1‍2α(0)kq∑j=-∞k-1‍λjq2(j-k)(s+δ+nδ2-α(0))q/2×∑j=-∞k-1‍2(j-k)(s+δ+nδ2-α(0))q′/2q/q′≲∑j=-∞-1‍2α(0)kq∑j=-∞k-1‍λjq2(j-k)(s+δ+nδ2-α(0))q/2=∑j=-∞-1‍λjq∑k=j+1-1‍2(j-k)(s+δ+nδ2-α(0))q/2≲Λ.So, we have II≲Λ.
Finally, we estimateIII: (89)III=sup⁡L>0,L∈Z2-Lλq∑k=0L‍2kqα∞(Tf)χkLp(·)q≲sup⁡L>0,L∈Z2-Lλq∑k=0L‍2kqα∞∑j=k∞‍λj(Tbj)χkLp(·)q+sup⁡L>0,L∈Z2-Lλq∑k=0L‍2kqα∞∑j=-∞k-1‍λj(Tbj)χkLp(·)q:=III1+III2.When 0<q⩽1, by the boundedness of T in Lp(·), we have (90)III1=sup⁡L>0,L∈Z2-Lλq∑k=0L‍2α∞kq∑j=k∞‍λj(Tbj)χkLp(·)q⩽sup⁡L>0,L∈Z2-Lλq∑k=0L‍2α∞kq∑j=k∞‍λjq(Tbj)χkLp(·)q≲sup⁡L>0,L∈Z2-Lλq∑k=0L‍2α∞kq∑j=k∞‍λjq2-αjjq≲sup⁡L>0,L∈Z2-Lλq∑k=0L‍2α∞kq∑j=k∞‍λjq2-α∞jq=sup⁡L>0,L∈Z2-Lλq∑j=0L‍λjq∑k=0j‍2(k-j)α∞q+sup⁡L>0,L∈Z2-Lλq∑j=L∞‍λjq∑k=0L‍2(k-j)α∞q≲sup⁡L>0,L∈Z2-Lλq∑j=0L‍λjq+sup⁡L>0,L∈Z∑j=L∞‍2(jλq-Lλq)2-jλq∑i=-∞j‍λiq∑k=0L‍2(k-j)α∞q≲Λ+Λsup⁡L>0,L∈Z∑j=L∞‍2(j-L)λq2(L-j)α∞q≲Λ+Λsup⁡L>0,L∈Z∑j=L∞‍2(j-L)q(λ-α∞)≲Λ.When 1<q<∞, by the boundedness of T in Lp(·) and Hölder’s inequality, we have (91)III1=sup⁡L>0,L∈Z2-Lλq∑k=0L‍2α∞kq∑j=k∞‍λj(Tbj)χkLp(·)q⩽sup⁡L>0,L∈Z2-Lλq∑k=0L‍2α∞kq∑j=k∞‍λjq(Tbj)χkLp(·)q/2×∑j=k∞‍(Tbj)χkLp(·)q′/2q/q′≲sup⁡L>0,L∈Z2-Lλq∑k=0L‍2α∞kq∑j=k∞‍λjqbjLp(·)q/2×∑j=k∞‍bjLp(·)q′/2q/q′≲sup⁡L>0,L∈Z2-Lλq∑k=0L‍2α∞kq∑j=k∞‍λjqBj-αjq/(2n)×∑j=k∞‍Bj-αjq′/(2n)q/q′≲sup⁡L>0,L∈Z2-Lλq∑k=0L‍2α∞kq/2∑j=k∞‍λjqBj-αjq/(2n)=sup⁡L>0,L∈Z2-Lλq∑j=0L‍λjq∑k=0j‍2(k-j)α∞q/2+sup⁡L>0,L∈Z2-Lλq∑j=L∞‍λjq∑k=0L‍2(k-j)α∞q/2≲sup⁡L>0,L∈Z2-Lλq∑j=0L‍λjq+sup⁡L>0,L∈Z∑j=L∞‍2(jλq-Lλq)2-jλq∑i=-∞j‍λiq∑k=0L‍2(k-j)α∞q/2≲Λ+Λsup⁡L>0,L∈Z∑j=L∞‍2(j-L)λq2(L-j)α∞q/2≲Λ+Λsup⁡L>0,L∈Z∑j=L∞‍2(j-L)q(λ-α∞/2)≲Λ.
When0<q⩽1, by nδ2⩽α(0), α∞<s+δ+nδ2 we get(92)III2=sup⁡L>0,L∈Z2-Lλq∑k=0L‍2α∞kq∑j=-∞k-1‍λj(Tbj)χkLp(·)q≲sup⁡L>0,L∈Z2-Lλq×∑k=0L‍2α∞kq∑j=-∞k-1‍λjq2[(s+δ+nδ2)(j-k)-jαj]q=sup⁡L>0,L∈Z2-Lλq×∑k=0L‍2α∞kq∑j=-∞-1‍λjq2[(s+δ+nδ2)(j-k)-jα(0)]q+sup⁡L>0,L∈Z2-Lλq×∑k=0L‍2α∞kq∑j=0k-1‍λjq2[(s+δ+nδ2)(j-k)-jα∞]q≲sup⁡L>0,L∈Z2-Lλq∑k=0L‍2α∞-s+δ+nδ2kq×∑j=-∞-1‍λjq2(s+δ+nδ2-α(0))jq+sup⁡L>0,L∈Z2-Lλq∑k=0L‍λjq∑k=j+1∞‍2(j-k)(s+δ+nδ2-α∞)q≲sup⁡L>0,L∈Z2-Lλq∑j=-∞-1‍λjq+sup⁡L>0,L∈Z2-Lλq∑j=0L-1‍λjq≲Λ.When 1<q<∞, let 1/q+1/q′=1. Since nδ2⩽α(0), α∞<s+δ+nδ2, by Hölder’s inequality, we have(93)III2=sup⁡L>0,L∈Z2-Lλq∑k=0L‍2α∞kq∑j=-∞k-1‍λj(Tbj)χkLp(·)q≲sup⁡L>0,L∈Z2-Lλq∑k=0L‍2α∞kq∑j=-∞k-1‍λj2[(s+δ+nδ2)(j-k)-jαj]q≲sup⁡L>0,L∈Z2-Lλq∑k=0L‍2α∞kq∑j=-∞-1‍λj2[(s+δ+nδ2)(j-k)-jα(0)]q+sup⁡L>0,L∈Z2-Lλq∑k=0L‍2α∞kq∑j=0k-1‍λj2[(s+δ+nδ2)(j-k)-jα∞]q≲sup⁡L>0,L∈Z2-Lλq∑k=0L‍2α∞-s+δ+nδ2kq×∑j=-∞-1‍λj2(s+δ+nδ2-α(0))jq+sup⁡L>0,L∈Z2-Lλq∑k=0L‍∑j=0k-1‍λj2(j-k)(s+δ+nδ2-α∞)q≲sup⁡L>0,L∈Z2-Lλq∑j=-∞-1‍λjq2(s+δ+nδ2-α(0))jq/2×∑j=-∞-1‍2(s+δ+nδ2-α(0))jq′/2q/q′+sup⁡L>0,L∈Z2-Lλq∑k=0L‍∑j=0k-1‍λjq2(j-k)(s+δ+nδ2-α∞)q/2×∑j=0k-1‍2(j-k)(s+δ+nδ2-α∞)q′/2q/q′≲sup⁡L>0,L∈Z2-Lλq∑j=-∞-1‍λjq2(s+δ+nδ2-α(0))jq/2+sup⁡L>0,L∈Z2-Lλq∑k=0L‍∑j=0k-1‍λjq2(j-k)(s+δ+nδ2-α∞)q/2≲sup⁡L>0,L∈Z2-Lλq∑j=-∞-1‍λjq+sup⁡L>0,L∈Z2-Lλq∑j=0L-1‍λjq∑k=j+1L‍2(j-k)(s+δ+nδ2-α∞)q/2≲sup⁡L>0,L∈Z2-Lλq∑j=-∞-1‍λjq+sup⁡L>0,L∈Z2-Lλq∑j=0L-1‍λjq≲Λ.Thus we have III≲Λ. This finishes the proof of Theorem 14.Definition 15.
LetK be a locally integrable function on Rn×Rn∖{x=y}. Then K is called a standard kernel if there exist δ∈(0,1] and C>0, such that(94)Kx,y⩽Cx-yn,x≠y;Kx,y-Kx,y′⩽Cy-y′δx-yn+δ,y-y′⩽12x-y;Kx,y-Kx′,y⩽Cx-x′δx-yn+δ,x-x′⩽12x-y.
A linear operatorT:S(Rn)→S′(Rn) is called a Calderón-Zygmund operator associated to a standard kernel K if(i)
T can be extended to a bounded operator on L2(Rn);(ii)
for anyf∈L2 with compact support and almost everywhere x∉suppf,(95)Tf(x)=∫Rn‍K(x,y)f(y)dy.It is well known that a Calderón-Zygmund operator is also bounded inLp(·) for any p(·)∈B(Rn); for example, see [38].Corollary 16.
Letα(·) be a bounded and log-Hölder continuous both at the origin and infinity such that nδ2⩽α(0), α∞<nδ2+δ with δ2 as in Lemma 4. Suppose T is a Calderón-Zygmund operator associated to a standard kernel K. If p(·)∈B(Rn), 0<q<∞, and 0≤λ<∞, then there exists a constant C such that(96)TfMK˙p(·),λα(·),q⩽CfHMK˙p·,λα·,q,TfMKp(·),λα(·),q⩽CfHMKp(·),λα(·),qfor f∈HMK˙p(·),λα(·),q and f∈HMKp(·),λα(·),q, respectively.Proof.
It is easy to know thats=max⁡{[α(0)-nδ2],[α∞-nδ2]}=0 when nδ2⩽α(0) and α∞<nδ2+δ. Then the result follows from Theorem 14.Remark 17.
For Hardy type spaces, there are some characterizations: maximal function, square function, atomic decomposition, and molecular decomposition. To discuss the boundedness of singular integrals in Hardy type spaces, we use the atomic decomposition for the domain Hardy space, while it is convenient to use another characterization of Hardy space for the target Hardy space; for example, see the proof of Theorem6.7.4 in [49] and [50, 51]. In a future paper, we will give the molecular decomposition of spaces HMK˙p(·),λα(·),q and HMKp(·),λα(·),q. Then, by the atomic and molecular decompositions, we will obtain the boundedness of T in Corollary 16 from HMK˙p(·),λα(·),q and HMKp(·),λα(·),q into themselves, respectively.



---

*Source: 160635-2015-04-06.xml*