Maximal operator on variable exponent spaces

TL;DR

This paper investigates the boundedness of the Hardy-Littlewood maximal operator on variable exponent spaces, extending classical results to unbounded exponents through new conditions and approximation techniques.

Contribution

It introduces a novel approach combining Muckenhoupt and Nekvinda conditions to establish boundedness for unbounded exponents in variable spaces.

Findings

Boundedness of M under combined conditions for unbounded exponents

Extension of classical boundedness results to variable exponent spaces

Introduction of an approximation method preserving key conditions

Abstract

We explore the boundedness of the Hardy-Littlewood maximal operator $M$ on variable exponent spaces. Our findings demonstrate that the Muckenhoupt condition, in conjunction with Nekvinda's decay condition, implies the boundedness of $M$ even for unbounded exponents. This extends the results of Lerner, Cruz-Uribe and Fiorenza for bounded exponents. We also introduce a novel argument that allows approximate unbounded exponents by bounded ones while preserving the Muckenhoupt and Nekvinda conditions.