KYUNGPOOK Math. J. 2019; 59(2): 259-275
On the Boundedness of Marcinkiewicz Integrals on Variable Exponent Herz-type Hardy Spaces
Rabah Heraiz
Department of Mathematics, Laboratory of Functional Analysis and Geometry of Spaces, M’sila University, P. O. Box 166, M’sila 28000, Algeria
e-mail : heraizrabeh@yahoo.fr and rabah.heraiz@univ-msila.dz
* Corresponding Author.
Received: March 22, 2018; Revised: November 2, 2018; Accepted: December 6, 2018; Published online: June 23, 2019.

This is an Open Access article distributed under the terms of the Creative Commons Attribution Non-Commercial License (http://creativecommons.org/licenses/by-nc/3.0/) which permits unrestricted non-commercial use, distribution, and reproduction in any medium, provided the original work is properly cited.
Abstract

The aim of this paper is to prove that Marcinkiewicz integral operators are bounded from $K˙p(·)α(·),q(·)(ℝn)$ to $K˙p(·)α(·),q(·)(ℝn)$ when the parameters α(·), p(·) and q(·) satisfies some conditions. Also, we prove the boundedness of μ on variable Herz-type Hardy spaces $HK˙p(·)α(·),q(·)(ℝn)$.

Keywords: Herz spaces, Herz-type Hardy spaces, variable exponent, HardyLittlewood maximal operator, Marcinkiewicz integral operators.
1. Introduction and Preliminaries

Function spaces with variable exponent are being actively studied not only in the field of real analysis but also in partial differential equations and in applied mathematics. The theory of function spaces with variable exponents has rapidly made progress in the last three decades.

For 0 < β ≤ 1, the Lipschitz space Lipβ(ℝn) is defined as

$Lipβ(ℝn):={f:‖f‖Lipβ(ℝn)=supx,y∈ℝn;x≠y∣f(x)-f(y)∣∣x-y∣β<∞}.$

Given Ω ∈ Lipβ(ℝn) be a homogeneous function of degree zero and

$∫Sn-1Ω(x′) dσ(x′)=0$

where x′ = x/|x| for any x ≠ 0 and Sn−1 denotes the unit sphere in ℝn (n ≥ 2) equipped with the normalized Lebesgue measure.

The Marcinkiewicz integral μ is defined by

$μ(f) (x):=(∫0∞∣FΩf(x)∣2dtt3)12$

where

$FΩf(x):=∫∣x-y∣≤tΩ(x-y)∣x-y∣n-1f(y) dy.$

It is well known that the operator μ was first defined by Stein [13] and under the conditions above, Stein proved that μ is of type (p, p) for 1 < p ≤ 2 and of weak type (1, 1). Benedek et al. [2] showed that μ of type (p, p) with 1 < p < ∞.

Recently, the boundedness of Marcinkiewicz integral operators μ on variable function spaces have attracted great attention (see [14, 15, 18] and their references).

The purpose of this paper is to generalize some results concerning Marcinkiewicz integral operators μ on variable Herz spaces $K˙p(·)α(·),q(·)(ℝn)$ and variable Herz-type Hardy spaces $HK˙p(·)α(·),q(·)(ℝn)$. We define the set of variable exponents by

$P0(ℝn):={p measurable:p(·):ℝn→[c,∞] for some c>0}.$

The subset of variable exponents with range [1,∞) is denoted by

2. Some Technical Lemmas
3. Variable Herz Estimate of Marcinkiewicz Integral Operators
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