Operators of Harmonic Analysis in Weighted Spaces with Non-standard Growth

TL;DR

This paper develops a variant of Rubio de Francia's extrapolation theorem for weighted variable exponent Lebesgue spaces and applies it to establish the boundedness of key harmonic analysis operators in these spaces.

Contribution

It introduces a new extrapolation theorem tailored for weighted variable exponent spaces and demonstrates its use in proving boundedness of multiple harmonic analysis operators.

Findings

Boundedness of maximal operators in weighted variable exponent spaces.

Boundedness of singular and potential operators in these spaces.

Extension to vector-valued operator analogues.

Abstract

Last years there was increasing an interest to the so called function spaces with non-standard growth, known also as variable exponent Lebesgue spaces. For weighted such spaces on homogeneous spaces, we develop a certain variant of Rubio de Francia's extrapolation theorem. This extrapolation theorem is applied to obtain the boundedness in such spaces of various operators of harmonic analysis, such as maximal and singular operators, potential operators, Fourier multipliers, dominants of partial sums of trigonometric Fourier series and others, in weighted Lebesgue spaces with variable exponent. There are also given their vector-valued analogues.