Boundedness of fractional integrals on weighted Herz spaces with variable exponent

Mitsuo Izuki,Takahiro Noi

doi:10.1186/s13660-016-1142-9

Mitsuo Izuki, Takahiro Noi

Open Access

https://doi.org/10.1186/s13660-016-1142-9

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Abstract

Our aim is to prove the boundedness of fractional integral operators on weighted Herz spaces with variable exponent. Our method is based on the theory on Banach function spaces and the Muckenhoupt theory with variable exponent.

Highlights

The boundedness of fractional integrals on function spaces is one of the important problems not in harmonic analysis and in potential theory and in partial differential equations
Among the development of variable exponent analysis the we can list up boundedness of fractional integrals on function spaces with variable exponent
The conditions on variable exponents have been established by the study of the boundedness of the Hardy-Littlewood maximal operator on spaces with variable exponent [ – ]

Summary

Introduction

The boundedness of fractional integrals on function spaces is one of the important problems not in harmonic analysis and in potential theory and in partial differential equations. Among the development of variable exponent analysis the we can list up boundedness of fractional integrals on function spaces with variable exponent. Cruz-Uribe, Fiorenza and Neugebauer [ ] and Diening and Hästö [ ] have independently proved the equivalence between the Muckenhoupt condition and the boundedness of the Hardy-Littlewood maximal operator on weighted Lebesgue spaces in the variable exponent setting. We note that CruzUribe and Wang [ ] have obtained the boundedness of fractional integrals on weighted Lebesgue spaces with variable exponent applying the extrapolation theorem. In this paper we define weighted Herz spaces with variable exponent and prove the boundedness of fractional integrals on those spaces under proper assumptions on weights and exponents. The authors have considered other problems on boundedness of some operators on weighted Herz spaces with variable exponent in the recent preprints [ , ].

Variable Lebesgue spaces

Herz spaces with variable exponent

The main results

Boundedness of fractional integrals on Herz spaces