Intrinsic Square Function Characterizations of Variable Hardy–Lorentz Spaces

Khedoudj Saibi

doi:10.1155/2020/2681719

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 intrinsicgλ∗-function of the variable Hardy–Lorentz spaceHp⋅,qℝn, forp⋅being a measurable function onℝnsatisfying0<p−≔ess infx∈ℝnpx≤ess supx∈ℝnpx≕p+<∞and the globally log-Hölder continuity condition andq∈0,∞, via its atomic and Littlewood–Paley function characterizations.

Highlights

In recent years, the theory of function spaces with variable exponents has gained great interest, see, for example, [1,2,3,4,5,6,7,8,9].e variable Lebesgue space is one of the generalizations of the classical Lp(Rn) space, originally introduced by Orlicz [10] via replacing p by the variable exponent function p(·): Rn ⟶ (0, ∞)
Under the assumption that the variable exponent p(·) satisfies the globally log-Holder 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
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. e Lorentz space Lp,q(Rn) tracked back to Lorentz [21] is another generalization of the classical Lp(Rn) space. is space forms a valuable topic in the theory of function spaces and harmonic analysis, see, for example, [5, 22,23,24,25,26]. e theory of Lorentz spaces was generalized to the Hardy–Lorentz space

Summary

Introduction

The theory of function spaces with variable exponents has gained great interest, see, for example, [1,2,3,4,5,6,7,8,9]. Under the assumption that the variable exponent p(·) satisfies the globally log-Holder 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. 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 c-order Calderon–Zygmund operators via the atomic characterization established in the same paper. Ho [35] broadened the mapping properties for intrinsic square functions to the weighted Hardy spaces with variable exponents studied in [20]. We use C and c to denote positive constants that are independent of the essential parameters involved but may differ from line to line. e symbol A ≲ B means A ≤ CB, and the symbol A ∼ B means A ≲ B and B ≲ A

Preliminaries

Findings

Intrinsic Square Function Characterizations