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.
© Kyungpook Mathematical Journal. All rights reserved.

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Abstract

The aim of this paper is to prove that Marcinkiewicz integral operators are bounded from K˙p(·)α(·),q(·)(n) to K˙p(·)α(·),q(·)(n) 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(·)(n).

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(n):={f:fLipβ(n)=supx,yn;xyf(x)-f(y)x-yβ<}.

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):=(0FΩf(x)2dtt3)12

where

FΩf(x):=x-ytΩ(x-y)x-yn-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(·)(n) and variable Herz-type Hardy spaces HK˙p(·)α(·),q(·)(n). We define the set of variable exponents by

P0(n):={pmeasurable:p(·):n[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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