Intrinsic Square Function Characterizations of Variable Hardy–Lorentz Spaces

Saibi, Khedoudj

doi:https://doi.org/10.1155/2020/2681719

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Research Article | Open Access

Volume 2020 | Article ID 2681719 | https://doi.org/10.1155/2020/2681719

Intrinsic Square Function Characterizations of Variable Hardy–Lorentz Spaces

Khedoudj Saibi¹

Academic Editor: Huy Qui Bui

Received31 Oct 2019

Revised09 Jan 2020

Accepted13 Jan 2020

Published10 Feb 2020

Abstract

The aim of this paper is to establish the intrinsic square function characterizations in terms of the intrinsic Littlewood–Paley -function, the intrinsic Lusin area function, and the intrinsic -function of the variable Hardy–Lorentz space , for being a measurable function on satisfying and the globally log-Hölder continuity condition and , via its atomic and Littlewood–Paley function characterizations.

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 space, originally introduced by Orlicz [10] via replacing by the variable exponent function . 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 spaces , introduced by Stein and Weiss in [16], is a well-known generalization of the Lebesgue spaces . This theory was also extended to the variable setting. To be more precise, under the assumption that the variable exponent satisfies the globally log-Hölder condition, Nakai and Sawano [17] introduced the variable Hardy space and established its atomic characterizations which were used to figure out its dual space and to prove the boundedness of singular integrals on 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 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 space tracked back to Lorentz [21] is another generalization of the classical 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 , Fefferman and Soria [28] studied and established its atomic characterization, Liu [29] established the atomic decomposition for with , and Abu-Shammala and Torchinsky [22] introduced the Hardy–Lorentz space , established its atomic characterizations, and proved the boundedness of singular integrals on for and . The classical Lorentz spaces were extended to the variable case, more precisely, Kempka and Vybíral [30] introduced the Lorentz spaces and proved that when is a constant which equals to q, the space is the real interpolation between and the variable Lebesgue space , and the authors have also showed that similar to a classical case that when 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 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 , constructed the atomic characterizations for this space, figured out its dual space, and proved the boundedness of singular integrals on .

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 function from the weighted Lebesgue space to , where and denotes the Hardy–Littlewood maximal function of , Wilson [31] introduced the intrinsic square functions and proved that they are bounded on the weighted Lebesgue with and the weight being in the Muckenhoupt class . The square functions of the form 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 space under the additional assumption that were established by Huang and Liu [36]. Systematically, Wang and Liu [37] have extended this result to for and , under other additional assumptions. Liang and Yang [38] established the s-order intrinsic square function characterization of the Musielak–Orlicz Hardy space , which was introduced by Ky [39] via -function and -function with the best range which improve the results obtained in [36, 38]. These results were extended to the variable Hardy space setting by [40] with . Recently, Yan [41] extended these results to the weak Musielak–Orlicz Hardy space with the same range as in [38], and the same author also established these characterizations for the variable weak Hardy space in [42].

Motivated by the series of papers [4, 8, 42], in this work, we aim to prove that the variable Hardy–Lorentz space 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. In Section 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 by and 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 means , and the symbol means and .

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 function . We denote by the collection of variable exponents satisfying where Let . The variable Lebesgue space is the set of measurable functions f such that equipped with the Luxemburg quasi-norm

In the next remark, we collect some basic properties of the variable Lebesgue spaces. For the proofs, see [12, 17].

Remark 1. Let and .(i)For , we have (ii)For we have (iii)If , thenhere and hereafter .
We recall the Fatou Lemma of obtained in [12], Theorem 2.61.

Lemma 1. Let and . If as pointwise almost everywhere in and is finite, then and

We say that the variable exponent satisfies the globally log-Hölder continuity condition, and we write if there exist constants , , and such that, for any

Let f be a locally integrable function and We recall that the Hardy–Littlewood maximal operator is defined bywhere the supremum is taken over all balls B of containing x.

Lemma 2. Let with . Then, there exists a positive constant C such that, for all see ([13], Theorem 4.3.8).

The vector-valued inequality for the Hardy–Littlewood maximal function on is given in the next lemma, (see [2], Corollary 2.1).

Lemma 3. Let and with . Then, there exists a positive constant C such that, for any sequence of measurable functions,

The next result can be easily proved using the fact that for any and the above lemma.

Corollary 1. Let , , and . Then, for any sequence , there exists a positive constant C such thatThe following technical lemma is a variant of [18], Theorem 4.1.

Lemma 4. Let , , and . Then, there exists a positive constant C such that, for all sequences of balls, numbers , and measurable functions satisfying that, for each , , and , it holds true that

2.2. Lorentz Spaces with Variable Exponents

We now recall the definition of Lorentz spaces with variable exponents considered in this paper.

Definition 1. Let and . Then, is defined to be the set of all measurable functions f such that whereLemma 5 presents an equivalent discrete characterization of the quasi-norm . For the proof, we refer to [30], Lemma 2.4.

Lemma 5. Let and . If , then

2.3. Variable Hardy–Lorentz Space and Existing Results

Denote by , the space of all Schwartz functions, and let denote its topological dual space. For , letwhere for any and

For all , define radial grand maximal function of f bywhere for all and , .

Definition 2. Let and be a positive integer. The Hardy–Lorentz space is defined to be the set of all such that , equipped with the quasi-normNext, we recall the definition of the atomic Hardy–Lorentz space given in [4], and before this, we recall the definition of -atom.

Definition 3. Let and . Fix an integer . A measurable function a on is called a -atom if there exists a ball B such that(1)supp (2)(3) for all with Let , be a sequence of numbers in , and be a sequence of balls in . Define

Definition 4. Let , , and s as in Definition 3. The variable atomic Hardy–Lorentz space is defined to be the space of all functions which can be decomposed aswhere is a sequence of -atoms, associated with balls , satisfying that, for all and with A being a positive constant independent of x and k and for all and , with C being a positive constant independent of k and j. Moreover, for any , we definewhere the infimum is taken over all the decompositions of f as (15).
The following result is Theorem 5.4 of [4].

Lemma 6. Let , and s as in Definition 3. Then, with equivalent quasi-norms.

Remark 2. Note that if , then there exists a sequence of -atoms, associated with balls , respectively, satisfying that, for any , such that f can be decomposed as in (15), where for all and , with C being a positive constant independent of k and j, andMoreover, by the definition of , the fact that for any , we haveLet be a radial function satisfyingRecall that the Littlewood–Paley g-function and the -function with are defined for all and byWe recall that a function is vanishing weakly at infinity if for any , in as . The following two propositions are just Theorem 8.2 and 8.3 in [4].

Proposition 1. Let and . Then, if and only if , f vanishes weakly at infinity, and . Moreover, for all ,where C is a positive constant independent of f.

Proposition 2. Let , , and . Then, if and only if , f vanishes weakly at infinity, and . Moreover, for all ,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.

Let denote the set of all functions with continuous classical derivatives up to an order less or equal to s by . For and , let denote the set of all functions satisfyingand for all and with ,

For all and , let

The intrinsic -function, the intrinsic Lusin area integral, and the intrinsic -function of f are defined, respectively, for all and by

For , , and , let denote the set of all functions satisfying for all , with , , , and for all and with ,

It is worth noting that the parameter ϵ has to be chosen large enough.

For all functions f such thatand , define

For all and , define

Remark 3. For , the intrinsic square functions were introduced by Wilson [31] and for by Liang and Yang in [38].
The next lemma is just [38], Proposition 2.4. For the case , it was proved in [31], p. 784.

Lemma 7. Let , , and . Then, for all f satisfying (28) and ,where the implicit positive constants are independent of f.

The next lemma is just [38], Theorem 2.6. For the case , it was proved in [31], p. 775.

Lemma 8. Let , , and . Then, there exists a positive constant C such that, for all f satisfying (28) and ,where the implicit positive constants are independent of f.

The following lemma is just [38], Proposition 3.2.

Lemma 9. Let , , and . Then, there exists a positive constant C such that, for all measurable function f,The main results of this work are stated as follows.

Theorem 1. Let with . Assume that , , and .(1)If , with denoting the dual space of , then there exists a positive constant C such that, for all , it holds true that(2)If , f vanishes weakly at infinity, and , then . Moreover, there exists a positive constant C such that

Theorem 2. Let with . Assume that , , , and .(1)If , with denoting the dual space of , then there exists a positive constant C such that, for all , it holds true that(2)If , f vanishes weakly at infinity, and , then . Moreover, there exists a positive constant C such that

Remark 4. (i) The same results in Theorem 1 hold true if we replace by , , and with .(ii)The same results in Theorem 2 hold true if we replace by with .

Proof of Theorem 1. By Lemmas 7 and 8, we know that, for , the functions , , , and are pointwise comparable, and we will prove the theorem just for .
We begin by proving (i). Let and . Note that since , , and , it follows that . Then, by Lemma 6 and Remark 2, there exists a sequence of -atoms, associated with balls , respectively, satisfying that, for any , and such that, for all and , with C being a positive constant independent of k and j such that f can be decomposed asFix . We write f asand let . Thus, we getNote thatWe firstly deal with . Let , , and , and by the Hölder inequality, we haveLet . Then, by Lemma 9, we know that, for all and ,Thus, following the argument used for (5.8) in [4], we can prove thatTo estimate , by [42], (2.7), we know that, for all and , we haveThus, for , , and , it was proved in [42] thatThen, by taking , the Hölder inequality for yields toTherefore,Thus, by (38), we haveNow, we pass to estimate . For this end, let and . Then, by Remark 1(ii), Lemmas 1 and 2.5, Remark 2, and the Hölder inequality, we obtainThen,Thus, by (38), we haveWe turn to estimate . Let such that (such that exists since ) and . Thus, by Remark 1(ii), Lemmas 1 and 2, and Remark 2, we getLet . Then, by (45), Lemma 3, and the Hölder inequality, we haveThen, following the argument used for (52), we can prove thatCombining the estimates (44), (49), (52), and (55), we getTo prove (ii), note that, for any , we haveBy this and the fact that vanishes weakly at infinity, , and Proposition 1, we deduce that andThe proof is complete.

Proof of Theorem 2. We begin by proving (i). Let . Following the argument used in [40], p. 1566, we can prove that, for any and ,Thus, by Theorem 1, we conclude thatWe turn to prove (ii). Let , f vanishes weakly at infinity, and . Note that since and , we haveBy this, the fact that, for all ,and Proposition 2, we deduce that andThis ends the proof.

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

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Copyright © 2020 Khedoudj Saibi. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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