Calderón Operator on Local Morrey Spaces with Variable Exponents

Kwok-Pun Ho

doi:10.3390/math9222977

Abstract

In this paper, we establish the boundedness of the Calderón operator on local Morrey spaces with variable exponents. We obtain our result by extending the extrapolation theory of Rubio de Francia to the local Morrey spaces with variable exponents. The exponent functions of the local Morrey spaces with the exponent functions are only required to satisfy the log-Hölder continuity assumption at the origin and infinity only. As special cases of the main result, we have Hardy’s inequalities, the Hilbert inequalities and the boundedness of the Riemann–Liouville and Weyl averaging operators on local Morrey spaces with variable exponents.

Highlights

IntroductionAs applications of our main results, we obtain the Hardy’ inequalities, the boundedness of the Stieltjes transformation, the Riemann–Liouville and Weyl averaging operators on local Morrey spaces with variable exponents
We see that whenever we can establish the weighted norm inequalities with the class of weight function A p,0 for an operator T, even if T is nonlinear, we can apply our extrapolation theory to obtain the boundedness of T on the local Morrey spaces with variable exponents where the exponent function is log-Hölder continuous at 0 and infinity
We extend the extrapolation theory to the local Morrey spaces with variable exponents with the exponent functions being log-Hölder continuous at the origin and infinity only

Summary

Introduction

As applications of our main results, we obtain the Hardy’ inequalities, the boundedness of the Stieltjes transformation, the Riemann–Liouville and Weyl averaging operators on local Morrey spaces with variable exponents

Definitions and Preliminaries

Local Morrey Spaces with Variable Exponents

Calderón Operator

Findings

Discussion

Conclusions