# Intrinsic Square Function Characterizations of Variable Hardy–Lorentz Spaces

Authors: Khedoudj Saibi
Journal: Journal of Function Spaces
 (2020)

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## Abstract

The aim of this paper is to establish the intrinsic square function characterizations in terms of the intrinsic Littlewood–Paleyg-function, the intrinsic Lusin area function, and the intrinsic gλ∗-function of the variable Hardy–Lorentz space Hp⋅,qℝn, for p⋅ being a measurable function on ℝn satisfying 0<p−≔essinfx∈ℝnpx≤esssupx∈ℝnpx≕p+<∞ and the globally log-Hölder continuity condition and q∈0,∞, via its atomic and Littlewood–Paley function characterizations.


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## Body

## 1. Introduction

In recent years, the theory of function spaces with variable exponents has gained great interest, see, for example, [1–9]. The variable Lebesgue space is one of the generalizations of the classical Lpℝn space, originally introduced by Orlicz [10] via replacing p by the variable exponent function p⋅:ℝn⟶0,∞. In 1991’s, Kováčik and Rákosník [11] proved some elementary properties for this kind of spaces. This space has been extensively studied by many researchers due to its wide use in different fields such as harmonic analysis and partial differential equations, see, for example, [12–15].The real variable theory of Hardy spacesHpℝn, introduced by Stein and Weiss in [16], is a well-known generalization of the Lebesgue spaces Lpℝn. This theory was also extended to the variable setting. To be more precise, under the assumption that the variable exponent p⋅ satisfies the globally log-Hölder condition, Nakai and Sawano [17] introduced the variable Hardy space Hp⋅ℝn and established its atomic characterizations which were used to figure out its dual space and to prove the boundedness of singular integrals on Hp⋅ℝn as well. Sawano [18] extended the atomic characterizations obtained in [17] and gave other applications of these kinds of Hardy spaces. In an independent way and under slightly weaker conditions to that used in [17, 18], Cruz-Uribe and Wang [19] also studied the variable Hardy space Hp⋅ℝn and constructed its atomic decomposition and showed the boundedness of several operators. Moreover, Ho [20] extended the atomic decompositions established in [17] to the weighted Hardy spaces with variable exponents and illustrated the relation between the boundedness of the Hardy–Littlewood maximal operators on function spaces and the atomic decompositions of Hardy-type spaces.The Lorentz spaceLp,qℝn tracked back to Lorentz [21] is another generalization of the classical Lpℝn space. This space forms a valuable topic in the theory of function spaces and harmonic analysis, see, for example, [5, 22–26]. The theory of Lorentz spaces was generalized to the Hardy–Lorentz space. Particularly, Fefferman et al. [27] investigated the real interpolation of the real Hardy–Lorentz space Hp,qℝn, Fefferman and Soria [28] studied H1,∞ℝn and established its atomic characterization, Liu [29] established the atomic decomposition for Hp,∞ with 0<p<1, and Abu-Shammala and Torchinsky [22] introduced the Hardy–Lorentz space Hp,qℝn, established its atomic characterizations, and proved the boundedness of singular integrals on Hp,qℝn for p∈0,1 and q∈0,∞. The classical Lorentz spaces Lp,qℝn were extended to the variable case, more precisely, Kempka and Vybíral [30] introduced the Lorentz spaces Lp⋅,q⋅ℝn and proved that when q⋅ is a constant which equals to q, the space Lp⋅,qℝn is the real interpolation between L∞ℝn and the variable Lebesgue space Lp⋅ℝn, and the authors have also showed that similar to a classical case that Lp⋅,q⋅ℝn=Lp⋅ℝn when p⋅=q⋅ and demonstrated that the Marcinkiewicz interpolation theorem does not work in the variable Lebesgue space setting.Recently, Yan et al. [8] introduced the variable weak Hardy space Hp⋅,∞ℝn by means of the radial grand maximal function and proved various characterizations including the atomic and molecular characterizations and investigated the boundedness of convolution δ-type and nonconvolution γ-order Caldéron–Zygmund operators via the atomic characterization established in the same paper. Very recently, Jiao et al. [4] investigated the variable Hardy–Lorentz space Hp⋅,qℝn, constructed the atomic characterizations for this space, figured out its dual space, and proved the boundedness of singular integrals on Hp⋅,qℝn.The study of the intrinsic square function on several function spaces has attracted steadily increasing interest. More precisely, in order to settle a conjecture proposed on the boundedness of the Lusin area functionSf from the weighted Lebesgue space Lℳv2ℝn to Lv2ℝn, where v∈Lloc1ℝn and ℳv denotes the Hardy–Littlewood maximal function of v, Wilson [31] introduced the intrinsic square functions and proved that they are bounded on the weighted Lebesgue Lwpℝn with p∈1,∞ and the weight w being in the Muckenhoupt class Apℝn. The square functions of the form Sf are all dominated by the intrinsic square functions; however, these later are not essentially bigger than any one of them. The generic nature of these functions make them pointwise equivalent to each other and extremely easy to work with, as Fefferman–Stein and Hardy–Littlewood maximal functions. Furthermore, the intrinsic Lusin area function has many advantages of being comparable at various opening cones which is a well-known property that does not hold for the classical Lusin area function, see [31–34]. Recently, Ho [35] broadened the mapping properties for intrinsic square functions to the weighted Hardy spaces with variable exponents studied in [20].Later, the intrinsic square function characterizations of the weighted Hardy spaceHw1ℝn under the additional assumption that f∈Lw1ℝn were established by Huang and Liu [36]. Systematically, Wang and Liu [37] have extended this result to Hwpℝn for p∈n/n+α,1 and α∈0,1, under other additional assumptions. Liang and Yang [38] established the s-order intrinsic square function characterization of the Musielak–Orlicz Hardy space Hφℝn, which was introduced by Ky [39] via g-function and gλ∗-function with the best range λ∈2+2α+s/n,∞ which improve the results obtained in [36, 38]. These results were extended to the variable Hardy space setting by [40] with λ∈3+2α+s,∞. Recently, Yan [41] extended these results to the weak Musielak–Orlicz Hardy space WHφℝn with the same range as in [38], and the same author also established these characterizations for the variable weak Hardy space Hp⋅,∞ℝn in [42].Motivated by the series of papers [4, 8, 42], in this work, we aim to prove that the variable Hardy–Lorentz space Hp⋅,qℝn can be characterized by means of the intrinsic square functions via using the atomic and the Littlewood–Paley function characterizations established in [4].We end this introduction by describing the sectionwise treatment of this article. InSection 2, we recall the definitions and existing results related to our work. Section 3 is devoted to establish the intrinsic square function characterization of the variable Hardy–Lorentz space.As usually, throughout the paper, we denote byℕ and ℤ the set of nonnegative integers and the set of integers, respectively. We use C and c to denote positive constants that are independent of the essential parameters involved but may differ from line to line. The symbol A≲B means A≤CB, and the symbol A∼B means A≲B and B≲A.


## 2. Preliminaries

In this section, we recall the definition and some properties of the variable Lebesgue space, Lorentz space with variable exponents, and the variable Hardy-Lorentz space, some lemmas, and existing results used in this work.


### 2.1. Variable Exponent Lebesgue Spaces

A variable exponent is a measurable functionp⋅:ℝn⟶0,∞. We denote by Pℝn the collection of variable exponents satisfying 0<p−≤p+<∞, where p−=essinfx∈ℝnpxandp+=esssupx∈ℝnpx. Let p⋅∈Pℝn. The variable Lebesgue space Lp⋅ℝn is the set of measurable functions f such that ∫ℝnfxpxdx<∞, equipped with the Luxemburg quasi-norm(1)fLp⋅ℝn=infλ>0:∫ℝnfxλpxdx≤1.In the next remark, we collect some basic properties of the variable Lebesgue spaces. For the proofs, see [12, 17].Remark 1.
Letp⋅∈Pℝn and f∈Lp⋅ℝn.(i)
Forλ∈ℂ, we have λfLp⋅ℝn=λfLp⋅ℝn.(ii)
Fors>0, we have fsLp⋅ℝn=fLsp⋅ℝns.(iii)
Ifg∈Lp⋅ℝn, then(2)f+gLp⋅ℝnp¯≤fLp⋅ℝnp¯+gLp⋅ℝnp¯,here and hereafter p¯=minp−,1.
We recall the Fatou Lemma ofLp⋅ℝn obtained in [12], Theorem 2.61.Lemma 1.
Letp⋅:ℝn⟶1,∞ and fkk∈ℕ⊂Lp⋅ℝn. If fk⟶f as k⟶∞ pointwise almost everywhere in ℝn and liminfk⟶∞fkLp⋅ℝn is finite, then f∈Lp⋅ℝn and(3)fLp⋅ℝn≤liminfk⟶∞fkLp⋅ℝn.We say that the variable exponentp⋅∈Pℝn satisfies the globally log-Hölder continuity condition, and we write p⋅∈Clogℝn if there exist constants Cp⋅, C∞, and p∞ such that, for any x,y∈ℝn,(4)px−py≤Cp⋅loge+1/x−y,px−p∞≤C∞loge+x.Letf be a locally integrable function and x∈ℝn. We recall that the Hardy–Littlewood maximal operator ℳ is defined by(5)ℳfx=supB∋x1B∫Bfydy,where the supremum is taken over all balls B of ℝn containing x.Lemma 2.
Letp⋅∈Clogℝn with 1<p−≤p+<∞. Then, there exists a positive constant C such that, for all f∈Lp⋅ℝn, ℳfLp⋅ℝn≤CfLp⋅ℝnsee ([13], Theorem 4.3.8).The vector-valued inequality for the Hardy–Littlewood maximal function onLp⋅ℝn is given in the next lemma, (see [2], Corollary 2.1).Lemma 3.
Letr∈1,∞ and p⋅∈Clogℝn with p−∈1,∞. Then, there exists a positive constant C such that, for any sequence fjj∈ℕ of measurable functions,(6)∑j=1∞ℳfjr1/rLp⋅ℝn≤C∑j=1∞fjr1/rLp⋅ℝn.The next result can be easily proved using the fact thatχκB≤κn/rℳχB1/r for any κ∈1,∞ and the above lemma.Corollary 1.
Letp⋅∈Clogℝn, τ∈0,min1,p−, and κ∈1,∞. Then, for any sequence Bjj∈ℕ, there exists a positive constant C such that(7)∑j∈ℕχκBjLp⋅ℝn≤Cκn/τ∑j∈ℕχBjLp⋅ℝn.
The following technical lemma is a variant of [18], Theorem 4.1.Lemma 4.
Letp⋅∈Clogℝn, r∈0,min1,p−, and q∈1,∞. Then, there exists a positive constant C such that, for all sequences Bjj∈ℕ of balls, numbers λjj∈ℕ⊂ℂ, and measurable functions ajj∈ℕ satisfying that, for each j∈ℕ, suppaj⊂Bj, and ajLqℝn≤Bj1/q, it holds true that(8)∑j∈ℕλjajr1/rLp⋅ℝn≤C∑j∈ℕλjχBjr1/rLp⋅ℝn.


### 2.2. Lorentz Spaces with Variable Exponents

We now recall the definition of Lorentz spaces with variable exponents considered in this paper.Definition 1.
Letp⋅∈Pℝn and 0<q≤∞. Then, Lp⋅,qℝn is defined to be the set of all measurable functions f such that fLp⋅,qℝn<∞, where(9)fLp⋅,qℝn=∫0∞λq−1χx∈ℝn:fx>λLp⋅ℝnqdλ1/q,if0<q<∞,supλ>0λχx∈ℝn:fx>λLp⋅ℝn,ifq=∞.
Lemma5 presents an equivalent discrete characterization of the quasi-norm ⋅Lp⋅,q. For the proof, we refer to [30], Lemma 2.4.Lemma 5.
Letp⋅∈Pℝn and 0<q≤∞. If f∈Lp⋅,qℝn, then(10)fLp⋅,qℝn∼∑k∈ℤ2kqχx∈ℝn:fx>2kLp⋅ℝnq1/q,if0<q<∞,supk∈ℤ2kχx∈ℝn:fx>2kLp⋅ℝn,ifq=∞.


### 2.3. Variable Hardy–Lorentz Space and Existing Results

Denote bySℝn, the space of all Schwartz functions, and let S′ℝn denote its topological dual space. For N∈ℕ, let(11)ℱNℝn≔ψ∈Sℝn:∑β∈ℤ+n,β≤Nsupx∈ℝn1+xNDβψx≤1,where for any β≔β1,…,βn∈ℤ+n,β=β1+⋯+βn and Dβ≔∂/∂x1β1,…,∂/∂xnβn.For allf∈S′ℝn, define radial grand maximal function fN,+∗ of f by(12)fN,+∗≔supf∗ψtx:t∈0,∞,ψ∈ℱNℝn,where for all t∈0,∞ and ξ∈ℝn, ψt≔t−nψξ/t.Definition 2.
Letp⋅∈Clogℝn and N∈n/p−+n+1,∞ be a positive integer. The Hardy–Lorentz space Hp⋅,qℝn is defined to be the set of all f∈S′ℝn such that fN,+∗∈Lp⋅,qℝn, equipped with the quasi-norm(13)fHp⋅,qℝn=fN,+∗Lp⋅,qℝn.
Next, we recall the definition of the atomic Hardy–Lorentz spaceHatom,r,sp⋅,qℝn given in [4], and before this, we recall the definition of p⋅,r,s-atom.Definition 3.
Letp⋅∈Pℝn and r>1. Fix an integer s∈n/p−−n−1,∞∩ℤ+. A measurable function a on ℝn is called a p⋅,r,s-atom if there exists a ball B such that(1)
suppa⊂B(2)
aLrℝn≤B1/r/χBLp⋅ℝn(3)
∫ℝnaxxαdx=0 for all α∈ℤ+ with α≤s
Letp⋅∈Pℝn, λk,jk∈ℤ,j∈ℕ be a sequence of numbers in ℂ, and Bk,jk∈ℤ,j∈ℕ be a sequence of balls in ℝn. Define(14)Aλk,jk∈ℤ,j∈ℕ,Bk,jk∈ℤ,j∈ℕ≔∑k∈ℤ∑j∈ℕλk,jχBk,jχBk,jLp⋅ℝnp¯1/p¯Lp⋅ℝnq1/q.Definition 4.
Letp⋅∈Pℝn, r∈1,∞, and s as in Definition 3. The variable atomic Hardy–Lorentz space Hatom,r,sp⋅,qℝn is defined to be the space of all functions f∈S′ℝn which can be decomposed as(15)f=∑k∈ℤ∑j∈ℕλk,jak,jinS′ℝn,where ak,jk∈ℤ,j∈ℕ is a sequence of p⋅,r,s-atoms, associated with balls Bk,jk∈ℤ,j∈ℕ, satisfying that, for all x∈ℝn and k∈ℤ,∑j∈ℕχBk,jx≤A with A being a positive constant independent of x and k and for all k∈ℤ and j∈ℕ, λk,j=C2kχBk,jLp⋅ℝn with C being a positive constant independent of k and j. Moreover, for any f∈Hatom,r,sp⋅,qℝn, we define(16)fHatom,r,sp⋅,qℝn≔infAλk,jk∈ℤ,j∈ℕ,Bk,jk∈ℤ,j∈ℕ,where the infimum is taken over all the decompositions of f as (15).
The following result is Theorem 5.4 of [4].Lemma 6.
Letp⋅∈Clogℝn,r∈maxp+,1,∞, and s as in Definition 3. Then, Hp⋅,qℝn=Hatom,r,sp⋅,qℝn with equivalent quasi-norms.Remark 2.
Note that iff∈Hatom,r,sp⋅,qℝn, then there exists a sequence ak,jk∈ℤ,j∈ℕ of p⋅,r,s-atoms, associated with balls Bk,jk∈ℤ,j∈ℕ, respectively, satisfying that, for any k∈ℤ, ∑j∈ℕχBk,jx≤A, such that f can be decomposed as in (15), where for all k∈ℤ and j∈ℕ, λk,j=C2kχBk,jLp⋅ℝn with C being a positive constant independent of k and j, and(17)fHatom,r,sp⋅,qℝn∼Aλk,jk∈ℤ,j∈ℕ,Bk,jk∈ℤ,j∈ℕ.
Moreover, by the definition ofλk,jk∈ℤ,j∈ℕ, the fact that for any k∈ℤ, ∑j∈ℕχBk,jx≤A, we have(18)fHatom,r,sp⋅,qℝn∼∑k∈ℤ2kq∑j∈ℕχBk,jLp⋅ℝnq1/q.
Letϕ∈Sℝn be a radial function satisfying(19)suppϕ⊂x∈ℝn:x≤1,∫ℝnϕxxγdx=0for allγ∈ℤ+withγ≤maxnp−−n−1,0,∫0∞ϕ^ξt2dtt=1for allξ∈ℝn\0⟶n.
Recall that the Littlewood–Paley g-function and thegλ∗-function with λ>0 are defined for all f∈S′ℝn and x∈ℝn by(20)gfx≔∫0∞f∗ϕtx2dtt1/2,gλ∗fx≔∫0∞∫ℝntt+x−yλnf∗ϕty2dtt1/2.
We recall that a functionf∈S′ℝn is vanishing weakly at infinity if for any ϕ∈Sℝn, f∗ϕt⟶0 in S′ℝn as t⟶∞. The following two propositions are just Theorem 8.2 and 8.3 in [4].Proposition 1.
Letp⋅∈Clogℝn and q∈0,∞. Then, f∈Hp⋅,qℝn if and only if f∈S′ℝn, f vanishes weakly at infinity, and gf∈Lp⋅,qℝn. Moreover, for all f∈Hp⋅,qℝn,(21)C−1gfLp⋅,qℝn≤fHp⋅,qℝn≤CgfLp⋅,qℝn,where C is a positive constant independent of f.Proposition 2.
Letp⋅∈Clogℝn, q∈0,∞, and λ∈1+2/minp−,2,∞. Then, f∈Hp⋅,qℝn if and only if f∈S′ℝn, f vanishes weakly at infinity, and gλ∗f∈Lp⋅,qℝn. Moreover, for all f∈Hp⋅,qℝn,(22)C−1gλ∗fLp⋅,qℝn≤fHp⋅,qℝn≤Cgλ∗fLp⋅,qℝn,where C is a positive constant independent of f.


## 2.1. Variable Exponent Lebesgue Spaces

A variable exponent is a measurable functionp⋅:ℝn⟶0,∞. We denote by Pℝn the collection of variable exponents satisfying 0<p−≤p+<∞, where p−=essinfx∈ℝnpxandp+=esssupx∈ℝnpx. Let p⋅∈Pℝn. The variable Lebesgue space Lp⋅ℝn is the set of measurable functions f such that ∫ℝnfxpxdx<∞, equipped with the Luxemburg quasi-norm(1)fLp⋅ℝn=infλ>0:∫ℝnfxλpxdx≤1.In the next remark, we collect some basic properties of the variable Lebesgue spaces. For the proofs, see [12, 17].Remark 1.
Letp⋅∈Pℝn and f∈Lp⋅ℝn.(i)
Forλ∈ℂ, we have λfLp⋅ℝn=λfLp⋅ℝn.(ii)
Fors>0, we have fsLp⋅ℝn=fLsp⋅ℝns.(iii)
Ifg∈Lp⋅ℝn, then(2)f+gLp⋅ℝnp¯≤fLp⋅ℝnp¯+gLp⋅ℝnp¯,here and hereafter p¯=minp−,1.
We recall the Fatou Lemma ofLp⋅ℝn obtained in [12], Theorem 2.61.Lemma 1.
Letp⋅:ℝn⟶1,∞ and fkk∈ℕ⊂Lp⋅ℝn. If fk⟶f as k⟶∞ pointwise almost everywhere in ℝn and liminfk⟶∞fkLp⋅ℝn is finite, then f∈Lp⋅ℝn and(3)fLp⋅ℝn≤liminfk⟶∞fkLp⋅ℝn.We say that the variable exponentp⋅∈Pℝn satisfies the globally log-Hölder continuity condition, and we write p⋅∈Clogℝn if there exist constants Cp⋅, C∞, and p∞ such that, for any x,y∈ℝn,(4)px−py≤Cp⋅loge+1/x−y,px−p∞≤C∞loge+x.Letf be a locally integrable function and x∈ℝn. We recall that the Hardy–Littlewood maximal operator ℳ is defined by(5)ℳfx=supB∋x1B∫Bfydy,where the supremum is taken over all balls B of ℝn containing x.Lemma 2.
Letp⋅∈Clogℝn with 1<p−≤p+<∞. Then, there exists a positive constant C such that, for all f∈Lp⋅ℝn, ℳfLp⋅ℝn≤CfLp⋅ℝnsee ([13], Theorem 4.3.8).The vector-valued inequality for the Hardy–Littlewood maximal function onLp⋅ℝn is given in the next lemma, (see [2], Corollary 2.1).Lemma 3.
Letr∈1,∞ and p⋅∈Clogℝn with p−∈1,∞. Then, there exists a positive constant C such that, for any sequence fjj∈ℕ of measurable functions,(6)∑j=1∞ℳfjr1/rLp⋅ℝn≤C∑j=1∞fjr1/rLp⋅ℝn.The next result can be easily proved using the fact thatχκB≤κn/rℳχB1/r for any κ∈1,∞ and the above lemma.Corollary 1.
Letp⋅∈Clogℝn, τ∈0,min1,p−, and κ∈1,∞. Then, for any sequence Bjj∈ℕ, there exists a positive constant C such that(7)∑j∈ℕχκBjLp⋅ℝn≤Cκn/τ∑j∈ℕχBjLp⋅ℝn.
The following technical lemma is a variant of [18], Theorem 4.1.Lemma 4.
Letp⋅∈Clogℝn, r∈0,min1,p−, and q∈1,∞. Then, there exists a positive constant C such that, for all sequences Bjj∈ℕ of balls, numbers λjj∈ℕ⊂ℂ, and measurable functions ajj∈ℕ satisfying that, for each j∈ℕ, suppaj⊂Bj, and ajLqℝn≤Bj1/q, it holds true that(8)∑j∈ℕλjajr1/rLp⋅ℝn≤C∑j∈ℕλjχBjr1/rLp⋅ℝn.


## 2.2. Lorentz Spaces with Variable Exponents

We now recall the definition of Lorentz spaces with variable exponents considered in this paper.Definition 1.
Letp⋅∈Pℝn and 0<q≤∞. Then, Lp⋅,qℝn is defined to be the set of all measurable functions f such that fLp⋅,qℝn<∞, where(9)fLp⋅,qℝn=∫0∞λq−1χx∈ℝn:fx>λLp⋅ℝnqdλ1/q,if0<q<∞,supλ>0λχx∈ℝn:fx>λLp⋅ℝn,ifq=∞.
Lemma5 presents an equivalent discrete characterization of the quasi-norm ⋅Lp⋅,q. For the proof, we refer to [30], Lemma 2.4.Lemma 5.
Letp⋅∈Pℝn and 0<q≤∞. If f∈Lp⋅,qℝn, then(10)fLp⋅,qℝn∼∑k∈ℤ2kqχx∈ℝn:fx>2kLp⋅ℝnq1/q,if0<q<∞,supk∈ℤ2kχx∈ℝn:fx>2kLp⋅ℝn,ifq=∞.


## 2.3. Variable Hardy–Lorentz Space and Existing Results

Denote bySℝn, the space of all Schwartz functions, and let S′ℝn denote its topological dual space. For N∈ℕ, let(11)ℱNℝn≔ψ∈Sℝn:∑β∈ℤ+n,β≤Nsupx∈ℝn1+xNDβψx≤1,where for any β≔β1,…,βn∈ℤ+n,β=β1+⋯+βn and Dβ≔∂/∂x1β1,…,∂/∂xnβn.For allf∈S′ℝn, define radial grand maximal function fN,+∗ of f by(12)fN,+∗≔supf∗ψtx:t∈0,∞,ψ∈ℱNℝn,where for all t∈0,∞ and ξ∈ℝn, ψt≔t−nψξ/t.Definition 2.
Letp⋅∈Clogℝn and N∈n/p−+n+1,∞ be a positive integer. The Hardy–Lorentz space Hp⋅,qℝn is defined to be the set of all f∈S′ℝn such that fN,+∗∈Lp⋅,qℝn, equipped with the quasi-norm(13)fHp⋅,qℝn=fN,+∗Lp⋅,qℝn.
Next, we recall the definition of the atomic Hardy–Lorentz spaceHatom,r,sp⋅,qℝn given in [4], and before this, we recall the definition of p⋅,r,s-atom.Definition 3.
Letp⋅∈Pℝn and r>1. Fix an integer s∈n/p−−n−1,∞∩ℤ+. A measurable function a on ℝn is called a p⋅,r,s-atom if there exists a ball B such that(1)
suppa⊂B(2)
aLrℝn≤B1/r/χBLp⋅ℝn(3)
∫ℝnaxxαdx=0 for all α∈ℤ+ with α≤s
Letp⋅∈Pℝn, λk,jk∈ℤ,j∈ℕ be a sequence of numbers in ℂ, and Bk,jk∈ℤ,j∈ℕ be a sequence of balls in ℝn. Define(14)Aλk,jk∈ℤ,j∈ℕ,Bk,jk∈ℤ,j∈ℕ≔∑k∈ℤ∑j∈ℕλk,jχBk,jχBk,jLp⋅ℝnp¯1/p¯Lp⋅ℝnq1/q.Definition 4.
Letp⋅∈Pℝn, r∈1,∞, and s as in Definition 3. The variable atomic Hardy–Lorentz space Hatom,r,sp⋅,qℝn is defined to be the space of all functions f∈S′ℝn which can be decomposed as(15)f=∑k∈ℤ∑j∈ℕλk,jak,jinS′ℝn,where ak,jk∈ℤ,j∈ℕ is a sequence of p⋅,r,s-atoms, associated with balls Bk,jk∈ℤ,j∈ℕ, satisfying that, for all x∈ℝn and k∈ℤ,∑j∈ℕχBk,jx≤A with A being a positive constant independent of x and k and for all k∈ℤ and j∈ℕ, λk,j=C2kχBk,jLp⋅ℝn with C being a positive constant independent of k and j. Moreover, for any f∈Hatom,r,sp⋅,qℝn, we define(16)fHatom,r,sp⋅,qℝn≔infAλk,jk∈ℤ,j∈ℕ,Bk,jk∈ℤ,j∈ℕ,where the infimum is taken over all the decompositions of f as (15).
The following result is Theorem 5.4 of [4].Lemma 6.
Letp⋅∈Clogℝn,r∈maxp+,1,∞, and s as in Definition 3. Then, Hp⋅,qℝn=Hatom,r,sp⋅,qℝn with equivalent quasi-norms.Remark 2.
Note that iff∈Hatom,r,sp⋅,qℝn, then there exists a sequence ak,jk∈ℤ,j∈ℕ of p⋅,r,s-atoms, associated with balls Bk,jk∈ℤ,j∈ℕ, respectively, satisfying that, for any k∈ℤ, ∑j∈ℕχBk,jx≤A, such that f can be decomposed as in (15), where for all k∈ℤ and j∈ℕ, λk,j=C2kχBk,jLp⋅ℝn with C being a positive constant independent of k and j, and(17)fHatom,r,sp⋅,qℝn∼Aλk,jk∈ℤ,j∈ℕ,Bk,jk∈ℤ,j∈ℕ.
Moreover, by the definition ofλk,jk∈ℤ,j∈ℕ, the fact that for any k∈ℤ, ∑j∈ℕχBk,jx≤A, we have(18)fHatom,r,sp⋅,qℝn∼∑k∈ℤ2kq∑j∈ℕχBk,jLp⋅ℝnq1/q.
Letϕ∈Sℝn be a radial function satisfying(19)suppϕ⊂x∈ℝn:x≤1,∫ℝnϕxxγdx=0for allγ∈ℤ+withγ≤maxnp−−n−1,0,∫0∞ϕ^ξt2dtt=1for allξ∈ℝn\0⟶n.
Recall that the Littlewood–Paley g-function and thegλ∗-function with λ>0 are defined for all f∈S′ℝn and x∈ℝn by(20)gfx≔∫0∞f∗ϕtx2dtt1/2,gλ∗fx≔∫0∞∫ℝntt+x−yλnf∗ϕty2dtt1/2.
We recall that a functionf∈S′ℝn is vanishing weakly at infinity if for any ϕ∈Sℝn, f∗ϕt⟶0 in S′ℝn as t⟶∞. The following two propositions are just Theorem 8.2 and 8.3 in [4].Proposition 1.
Letp⋅∈Clogℝn and q∈0,∞. Then, f∈Hp⋅,qℝn if and only if f∈S′ℝn, f vanishes weakly at infinity, and gf∈Lp⋅,qℝn. Moreover, for all f∈Hp⋅,qℝn,(21)C−1gfLp⋅,qℝn≤fHp⋅,qℝn≤CgfLp⋅,qℝn,where C is a positive constant independent of f.Proposition 2.
Letp⋅∈Clogℝn, q∈0,∞, and λ∈1+2/minp−,2,∞. Then, f∈Hp⋅,qℝn if and only if f∈S′ℝn, f vanishes weakly at infinity, and gλ∗f∈Lp⋅,qℝn. Moreover, for all f∈Hp⋅,qℝn,(22)C−1gλ∗fLp⋅,qℝn≤fHp⋅,qℝn≤Cgλ∗fLp⋅,qℝn,where C is a positive constant independent of f.


## 3. Intrinsic Square Function Characterizations

In this section, we prove that the variable Hardy–Lorentz space can be characterized by means of the intrinsic square functions. We firstly begin by introducing some terminologies.Lets∈ℤ+ denote the set of all functions with continuous classical derivatives up to an order less or equal to s by Csℝn. For α∈0,1 and s∈ℤ+, let Cα,sℝn denote the set of all functions ϕ∈Csℝn satisfying(23)suppϕ⊂x∈ℝn:x≤1,∫ℝnϕxxγdx=0for allγ∈ℤ+nwithγ≤s,and for all x1,x2∈ℝn and ν∈ℤ+n with ν=s,(24)Dνϕx1−Dνϕx2≤x1−x2α.For allf∈Lloc1ℝn and y,t∈ℝn×0,∞, let(25)Aα,sfy,t≔supϕ∈Cα,sℝnf∗ϕty.The intrinsicg-function, the intrinsic Lusin area integral, and the intrinsic gλ∗-function of f are defined, respectively, for all x∈ℝn and λ>0 by(26)gα,sfx≔∫0∞Aα,sfx,t2dtt1/2,Sα,sfx≔∫ΓxAα,sfx,t2dttn+11/2,gλ,α,s∗fx≔∫0∞∫ℝntt+x−yλnAα,sfx,t2dtt1/2.Forα∈0,1, s∈ℤ+, and ϵ∈0,∞, let Cα,ε,sℝn denote the set of all functions ϕ∈Csℝn satisfying for all x∈ℝn, γ∈ℤ+n with γ≤s, Dγϕx≤1+x−n−ε, ∫ℝnϕxxγdx=0, and for all x1,x2∈ℝn and ν∈ℤ+n with ν=s,(27)Dνϕx1−Dνϕx2≤x1−x2α1+x1−n−ε+1+x2−n−ε.It is worth noting that the parameterϵ has to be chosen large enough.For all functionsf such that(28)f⋅1+⋅−n−ε∈L1ℝnand y,t∈ℝn×0,∞, define(29)A˜α,ε,sy,t≔supϕ∈Cα,ε,sℝnf∗ϕty.For allx∈ℝn and λ>0, define(30)g˜α,ε,sfx≔∫0∞A˜α,ε,sfx,t2dtt1/2,S˜α,ε,sfx≔∫ΓxA˜α,ε,sfx,t2dttn+11/2,g˜λ,α,s∗fx≔∫0∞∫ℝntt+x−yλnA˜α,ε,sfx,t2dtt1/2.Remark 3.
Fors=0, the intrinsic square functions were introduced by Wilson [31] and for s∈ℤ+ by Liang and Yang in [38].
The next lemma is just [38], Proposition 2.4. For the case s=0, it was proved in [31], p. 784.Lemma 7.
Letα∈0,1, s∈ℤ+, and ε∈0,∞. Then, for all f satisfying (28) and x∈ℝn,(31)gα,sfx∼Sα,sfx,g˜α,ε,sfx∼S˜α,ε,sfx,where the implicit positive constants are independent of f.The next lemma is just [38], Theorem 2.6. For the case s=0, it was proved in [31], p. 775.Lemma 8.
Letα∈0,1, s∈ℤ+, and ε∈maxα,s,∞. Then, there exists a positive constant C such that, for all f satisfying (28) and x∈ℝn,(32)C−1gα,sfx≤g˜α,ε,sfx≤Cgα,sfx,where the implicit positive constants are independent of f.The following lemma is just [38], Proposition 3.2.Lemma 9.
Letα∈0,1, s∈ℤ+, and q∈1,∞. Then, there exists a positive constant C such that, for all measurable function f,(33)∫ℝngα,sfxqdx≤C∫ℝnfxqdx.
The main results of this work are stated as follows.Theorem 1.
Letp⋅∈Clogℝn with p+∈0,1. Assume that α∈0,1, s∈ℤ+, and p−∈n/n+α+s,1.(1)
Iff∈Cα,ε,sℝn∗, with Cα,ε,sℝn∗ denoting the dual space of Cα,ε,sℝn, then there exists a positive constant C such that, for all f∈Hp⋅,qℝn, it holds true that(34)gα,sfLp⋅,qℝn≤CfHp⋅,qℝn.(2)
Iff∈S′ℝn, f vanishes weakly at infinity, and gα,sf∈Lp⋅,qℝn, then f∈Hp⋅,qℝn. Moreover, there exists a positive constant C such that(35)fHp⋅,qℝn≤Cgα,sfLp⋅,qℝn.Theorem 2.
Letp⋅∈Clogℝn with p+∈0,1. Assume that α∈0,1, s∈ℤ+, p−∈n/n+α+s,1, and λ∈3+2α+s/n,∞.(1)
Iff∈Cα,ε,sℝn∗, with Cα,ε,sℝn∗ denoting the dual space of Cα,ε,sℝn, then there exists a positive constant C such that, for all f∈Hp⋅,qℝn, it holds true that(36)gλ,α,s∗fLp⋅,qℝn≤CfHp⋅,qℝn.(2)
Iff∈S′ℝn, f vanishes weakly at infinity, and gα,sf∈Lp⋅,qℝn, then f∈Hp⋅,qℝn. Moreover, there exists a positive constant C such that(37)fHp⋅,qℝn≤Cgλ,α,s∗fLp⋅,qℝn.Remark 4.
(i) The same results in Theorem1 hold true if we replace gα,s by Sα,s, g˜α,s, and S˜α,s with ε∈maxα,s,∞.(ii)
The same results in Theorem2 hold true if we replace gλ,α,s∗ by g˜λ,α,ε,s with ε∈maxα,s,∞.Proof of Theorem1.
By Lemmas7 and 8, we know that, for ε∈maxα,s,∞, the functions gα,s, Sα,s, g˜α,s, and S˜α,s are pointwise comparable, and we will prove the theorem just for gα,s.
We begin by proving (i). Letf∈Hp⋅,qℝn and r∈1,∞. Note that since α∈0,1, s∈ℤ+, and p−∈n/n+α+s,1, it follows that s∈n/p−−n−1,∞∩ℤ+. Then, by Lemma 6 and Remark 2, there exists a sequence ak,jk∈ℤ,j∈ℕ of p⋅,r,s-atoms, associated with balls Bk,jk∈ℤ,j∈ℕ, respectively, satisfying that, for any k∈ℤ, ∑j∈ℕχBk,jx≤A, and λk,jk∈ℤ,j∈ℕ⊂ℂ such that, for all k∈ℤ and j∈ℕ, λk,j=C2kχBk,jLp⋅ℝn with C being a positive constant independent of k and j such that f can be decomposed as(38)f=∑k∈ℤ∑j∈ℕλk,jak,jinS′ℝn,fHp⋅,qℝn∼fHatom,r,sp⋅,qℝn∼∑k∈ℤ2kq∑j∈ℕχBk,jLp⋅ℝnq1/q.
Fixk0∈ℤ. We write f as(39)f=∑k=−∞k0−1∑j∈ℕλk,jak,j+∑k=k0∞∑j∈ℕλk,jak,j≕f1+f2,and let Ak0=∪k=k0∞∪j∈ℕ2Bk,j. Thus, we get(40)χx∈ℝn:gα,sfx>2k0Lp⋅ℝn≲χx∈ℝn:gα,sf1x>2k0−1Lp⋅ℝn+χx∈Ak0:gα,sf2x>2k0−1Lp⋅ℝn+χx∈Ak0c:gα,sf2x>2k0−1Lp⋅ℝn≕I1+I2+I3.
Note that(41)I1≲χx∈ℝn:∑k=−∞k0−1∑j∈ℕλk,jgα,sak,jχ2Bk,jx>2k0−2Lp⋅ℝn+χx∈ℝn:∑k=−∞k0−1∑j∈ℕλk,jgα,sak,jχ2Bk,jcx>2k0−2Lp⋅ℝn≔J1+J2.
We firstly deal withJ1. Let β1∈0,minp−,q, η1∈1,minr,1/β1, and σ1∈0,1−1/η1, and by the Hölder inequality, we have(42)∑k=−∞k0−1∑j∈ℕλk,jgα,sak,j≤∑k=−∞k0−12kσ1η1′1/η1′∑k=−∞k0−12−kσ1η1∑j∈ℕλk,jgα,sak,jη11/η1.
Letv=r/η1. Then, by Lemma 9, we know that, for all k∈ℤ and j∈ℕ,(43)χBk,jLp⋅ℝngα,sak,jη1χ2Bk,jLvℝn≲Bk,j1/v.
Thus, following the argument used for (5.8) in [4], we can prove that(44)∑k0∈ℤ2k0qJ1q1/q≲fHatom,r,sp⋅,qℝn.
To estimateJ2, by [42], (2.7), we know that, for all k∈ℤ and j∈ℕ, we have(45)gα,sak,jx≲χBk,jLp⋅ℝn−1MχBk,jn+α+s/n.
Thus, forβ2∈0,minn/n+α+s,q, η2∈n/n+α+sβ2,1/β2, and σ2∈0,1−1/η2, it was proved in [42] that(46)J2≲2−k0η21−σ2∑k=−∞k0−121−σ2kη2β2∑j∈ℕχBk,jLp⋅/β2ℝn1/β2=2−k0η21−σ2∑k=−∞k0−121−σ2kη2β2∑j∈ℕχBk,j1/β2Lp⋅ℝnβ21/β2.
Then, by takingγ2∈1,1−σ2η2, the Hölder inequality for q−β2/q+β2/q=1 yields to(47)J2≲2−k0η21−σ2∑k=−∞k0−121−σ2kη2β2−kγ2β2+kγ2β2∑j∈ℕχBk,j1/β2Lp⋅ℝnβ21/β2≲2−k0γ2∑k=−∞k0−12kqγ2∑j∈ℕχBk,jLp⋅ℝnq1/q.
Therefore,(48)∑k0∈ℤ2k0qJ2q≲∑k0∈ℤ2k0q2−k0γ2q∑k=−∞k0−12kqγ2∑j∈ℕχBk,jLp⋅ℝnq=∑k=−∞∞2kqγ2∑j∈ℕχBk,jLp⋅ℝnq∑k0=k+1∞2k0q1−γ2≲∑k∈ℤ2kq∑j∈ℕχBk,jLp⋅ℝnq.
Thus, by (38), we have(49)∑k0∈ℤ2k0qJ2q1/q≲fHatom,r,sp⋅,qℝn.
Now, we pass to estimateI2. For this end, let β3∈0,minp−,q and γ3∈0,1. Then, by Remark 1(ii), Lemmas 1 and 2.5, Remark 2, and the Hölder inequality, we obtain(50)I2≤χAk0Lp⋅ℝn≤∑k=k0∞∑j∈ℕχ2Bk,jLp⋅ℝn≲∑k=k0∞∑j∈ℕχBk,jLp⋅ℝn≲∑k=k0∞∑j∈ℕχBk,jLp⋅ℝnβ31/β3=∑k=k0∞2kγ3β32−kγ3β3∑j∈ℕχBk,jLp⋅ℝnβ31/β3≲∑k=k0∞2kγ3q∑j∈ℕχBk,jLp⋅ℝnq1/q∑k=k0∞2−kγ3β3q/q−β3q−β3/qβ3≲2−k0γ3∑k=k0∞2kγ3q∑j∈ℕχBk,jLp⋅ℝnq1/q.
Then,(51)∑k0=−∞∞2k0I2q≲∑k0=−∞∞2k0q2−k0γ3q∑k=k0∞2kqγ3∑j∈ℕχBk,jLp⋅ℝnq≲∑k=−∞∞2kqγ3∑j∈ℕχBk,jLp⋅ℝnq∑k0=−∞k2k0q1−γ3≲∑k=−∞∞2kq∑j∈ℕχBk,jLp⋅ℝnq.
Thus, by (38), we have(52)∑k0∈ℤ2k0qI2q1/q≲fHatom,r,sp⋅,qℝn.
We turn to estimateI3. Let σ4∈0,∞ such that σ4∈n/p−n+α+s,1 (such that σ4 exists since p−∈n/n+α+s,1) and β4∈0,minn/σ4n+α+s,q. Thus, by Remark 1(ii), Lemmas 1 and 2, and Remark 2, we get(53)I3≲2−k0σ4∑k=k0∞∑j∈ℕλk,jgα,sak,jσ4χAk0cLp⋅ℝn≲2−k0σ4∑k=k0∞∑j∈ℕλk,jgα,sak,jσ4χAk0cβ4Lp⋅/β4ℝn1/β4.
Letγ4∈σ4,1. Then, by (45), Lemma 3, and the Hölder inequality, we have(54)I3≲2−k0σ4∑k=k0∞2kσ4β4∑j∈ℕMχBk,j1/β4β4Lp⋅/β4ℝn1/β4≲2−k0σ4∑k=k0∞2kσ4β4∑j∈ℕχBk,jLp⋅ℝnβ41/β4≲2−k0σ4∑k=k0∞2kσ4β42−kγ4β42kγ4β4∑j∈ℕχBk,jLp⋅ℝnβ41/β4≲2−kσ4∑k=k0∞2kβ4σ4−γ4q/q−β4q−β4/qβ4∑k=k0∞2kγ4q∑j∈ℕχBk,jLp⋅ℝnq1/q≲2−k0γ4∑k=k0∞2kγ4q∑j∈ℕχBk,jLp⋅ℝnq1/q.
Then, following the argument used for (52), we can prove that(55)∑k0∈ℤ2k0qI3q1/q≲fHatom,r,sp⋅,qℝn.
Combining the estimates (44), (49), (52), and (55), we get(56)gα,sfLp⋅,qℝn≤CfHatom,r,sp⋅,qℝn∼fHp⋅,qℝn.
To prove (ii), note that, for anyx∈ℝn, we have(57)gfx≲gα,sfx.
By this and the fact thatf∈S′ℝn vanishes weakly at infinity, gα,sf∈Lp⋅,qℝn, and Proposition 1, we deduce that f∈Hp⋅,qℝn and(58)fHp⋅,qℝn≲gfLp⋅,qℝn≲gα,sfLp⋅,qℝn.
The proof is complete.Proof of Theorem 2.
We begin by proving (i). Letf∈Hp⋅,qℝn. Following the argument used in [40], p. 1566, we can prove that, for any α∈0,1,s∈ℤ+,ε∈maxα,s,∞,λ∈3+2α+s/n,∞ and x∈ℝn,(59)g˜λ,α,ε,s∗fx≲S˜α,ε,sfx.
Thus, by Theorem1, we conclude that(60)gλ,α,s∗fLp⋅,qℝn≲g˜λ,α,ε,s∗fLp⋅,qℝn≲S˜α,ε,sfLp⋅,qℝn≲fHp⋅,qℝn.
We turn to prove (ii). Letf∈S′ℝn, f vanishes weakly at infinity, and gλ,α,s∗f∈Lp⋅,qℝn. Note that since α∈0,1,s∈ℤ+,p−∈n/n+α+s,1 and λ∈3+2α+s/n,∞, we have(61)λ>1+2minp−,2.
By this, the fact that, for allx∈ℝn,(62)gλ∗fx≲gλ,α,s∗fx≲g˜λ,α,ε,s∗fx,and Proposition 2, we deduce that f∈Hp⋅,qℝn and(63)fHp⋅,qℝn≲g∗fLp⋅,qℝn≲gλ,α,s∗fLp⋅,qℝn≲g∗λ,α,ε,sfLp⋅,qℝn.
This ends the proof.



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