Local Hardy-Littlewood maximal opeator in variable Lebesgue spaces

TL;DR

This paper studies the boundedness of the local Hardy-Littlewood maximal operator in variable Lebesgue spaces and provides a Littlewood-Paley square-function characterization for these spaces.

Contribution

It introduces a class of exponents for which the local maximal operator is bounded and characterizes variable Lebesgue spaces using Littlewood-Paley theory.

Findings

Boundedness of local Hardy-Littlewood maximal operator in specified variable Lebesgue spaces.

Littlewood-Paley square-function characterization of these spaces.

Identification of a new class of exponents for boundedness.

Abstract

We investigate the class $\mathcal{B}^{loc}(\mathbb{R}^{n})$ of exponents $p(\cdot)$ for with local Hardy-Littlewood maximal operator is bounded in $L^{p(\cdot)}(\mathbb{R}^{n})$ space. Littlewood-Paley square-function characterization of $L^{p(\cdot)}(\mathbb{R}^{n})$ spaces with the above class of exponent are also obtained.