Abstract
The aim of this paper is to study the boundedness of Calderón-Zygmund operator and their commutator on Herz Spaces with two variable exponents p(.),q(.). By applying the properties of the Lebesgue spaces with variable exponent, the boundedness of the Calderón-Zygmund operator and the commutator generated by BMO function and Calderón-Zygmund operator is obtained on Herz space.
Highlights
A standard operator T is called a γ -Calderón - Zygmund operator if K is a standard kernel satisfies:
The function spaces with variable exponent has been recently obtained an increasing interest by a number of authors since many applications are found in many different fields, for example, in fluid dynamics, image restoration and differential equations
Herz spaces play an important role in harmonic analysis
Summary
S′ n (see [1], [2]). T is called a standard operator if T satisfies the following conditions:. The commutator of the Calderón-Zygmund operator is defined by [= b,T ] f ( x) b ( x)Tf ( x) − T (bf )( x). Jouné proved γ -Calderón - Zygmund operator is bounded on ( ) Lp n in [3]. Herz spaces play an important role in harmonic analysis. After they were introduced in [10], the boundedness of some operators and some characterizations of Herz spaces with variable exponents were studied extensively (see [11]-[16]). We will discuss the boundedness of the Calderón-Zygmund operator T and their commutator [b,T ] are bounded on Herz spaces with two variable exponents p (.), q (.)
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