Boundedness of Calderón-Zygmund operators and their commutator on Morrey-Herz Spaces with variable exponents

Omer Abdalrhman,Afif Abdalmonem,Shuangping Tao

doi:10.30538/oms2020.0122

Abstract

Boundedness of Calderón-Zygmund operators and their commutator on Morrey-Herz Spaces with variable exponents

Highlights

L et K be a locally integrable function on Rn × Rn\{(x, y) : x = y}, we say that K is a standard kernel if there exist ε > 0 and C > 0, such that|K(x, y)| ≤ C/|x − y|n, x = y; |y − w|ε|K(x, y) − K(x, w)| ≤ C |x − y|n+ε, |y − w| ≤ 2 |x − y|; |x − z|ε|K(x, y) − K(z, y)| ≤ C |x − y|n+ε, |x − z| ≤ 2 |x − y|.We say that a linear operator T : S (Rn) −→ S (Rn) is a Calderón−Zygmund operator associated to a standard kernel K if1
In this paper, the boundedness of Calderón−Zygmund operators is obtained on Morrey-Herz spaces with variable exponents MKqα((··)),pλ(·)(Rn) and several norm inequalities for the commutator generated by Calderón−Zygmund operators, BMO function and Lipschitz function are given
Et K be a locally integrable function on Rn × Rn\{(x, y) : x = y}, we say that K is a standard kernel if there exist ε > 0 and C > 0, such that

Summary

Introduction

One of the main problems on the theory of function spaces is the boundedness of the Hardy-Littlewood maximal operator on Lebesgue spaces with variable exponent. Many researchers [6–9] considered the question of sufficient conditions on the exponent function p(x) to obtained the boundedness of Hardy-Littlewood maximal operators. Jouné proved that if T is a ε-Calderón−Zygmund operator, T is bounded on Lp(Rn) [10]. Showed the commutator [b, T on Herz-Type spaces. In 2006, Cruz-Uribe et al, [13] established the boundedness of some classical operators on variable Lp spaces by applying the theory of weighed norm inequalities and extrapolation. The boundedness of some operators and their corresponding characterization of these spaces with variable exponent p(x) were studied widely [15,16]. Morrey-Herz spaces MKqα((··)),,pλ(·)(Rn) and MKqα((··)),,pλ(·)(Rn) with three variable exponents were studied by Wang and Tao [17]

Definition of function spaces with variable exponent

Main result and proof