Abstract
We define the new central Morrey space with variable exponent and investigate its relation to the Morrey-Herz spaces with variable exponent. As applications, we obtain the boundedness of the homogeneous fractional integral operator TΩ,σ and its commutator [b,TΩ,σ] on Morrey-Herz space with variable exponent, where Ω∈Ls(Sn-1) for s≥1 is a homogeneous function of degree zero, 0<σ<n, and b is a BMO function.
Highlights
The Morrey spaces first appeared in 1938 in the work of Morrey [1] in relation to some problems in partial differential equations
The Herz spaces are a class of function spaces introduced by Herz in the study of absolutely convergent Fourier transforms in 1968; see [3]
The complete theory of Herz spaces for the case of general indexes was established by Lu et al in 2008; see [4]
Summary
The Morrey spaces first appeared in 1938 in the work of Morrey [1] in relation to some problems in partial differential equations. The theory of function spaces with variable exponent was extensively studied by researchers since the work of Kovacik and Rakosnık [6] appeared in 1991; see [7, 8] and the references therein Many applications of these spaces were given, for example, in the modeling of electrorheological fluids [9], in the study of image processing [10], and in differential equations with nonstandard growth [11]. Motivated by the above references, in the present paper we will study the boundedness for the homogeneous fractional integral operator TΩ,σ and its commutator [b, TΩ,σ] on the Morrey-Herz space with variable exponent. We will establish the boundedness for the homogeneous fractional integral operator TΩ,σ and its commutator [b, TΩ,σ] on the MorreyHerz space with variable exponent in Sections 4 and 5
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