Extensions of Rubio de Francia's extrapolation theorem in variable Lebesgue space and application

TL;DR

This paper extends Rubio de Francia's extrapolation theorem to variable Lebesgue spaces, enabling new boundedness results for various singular and maximal operators in these spaces.

Contribution

It introduces a variant of the extrapolation theorem for variable exponent Lebesgue spaces, broadening the scope of operator boundedness results.

Findings

Boundedness of strongly singular integral operators established

Boundedness of singular integral operators with rough kernels demonstrated

Boundedness of fractional maximal and Bochner-Riesz operators shown

Abstract

We obtain one variant of the extrapolation theorem of Rubio de Fracia for variable exponent Lebesgue spaces. As a consequence we obtain conditions guarantee boundedness of strongly singular integral operators, singular integral operators with rough kernels, fractional maximal operators related to spherical means, Bochner-Riesz operators in variable Lebesgue spaces.