Wavelet characterization of local Muckenhoupt weighted Lebesgue spaces with variable exponent

TL;DR

This paper characterizes local Muckenhoupt weighted Lebesgue spaces with variable exponents using wavelets and establishes conditions for modular inequalities, including handling weights with exponential growth.

Contribution

It introduces a wavelet-based characterization for these spaces and provides necessary and sufficient conditions for modular inequalities, extending to exponential growth weights.

Findings

Wavelet characterization of variable exponent spaces

Necessary and sufficient conditions for modular inequalities

Handling of exponential growth weights

Abstract

Our aim in this paper is to characterize local Muckenhoupt weighted Lebesgue spaces with variable exponent by compactly supported smooth wavelets. We also investigate necessary and sufficient conditions for the corresponding modular inequalities to hold. One big achievement is that the weights with exponetial growth can be handled in the framework of variable exponents.