Sufficient condition for boundedness of maximal operator on weighted generalized Orlicz spaces

TL;DR

This paper establishes conditions under which the Hardy-Littlewood maximal operator remains bounded in weighted generalized Orlicz spaces, extending classical results by incorporating specific growth and weight conditions.

Contribution

It introduces new sufficient conditions involving Muckenhoupt weights and growth conditions for the boundedness of the maximal operator in these spaces.

Findings

Boundedness of maximal operator under Muckenhoupt $A_p$ weights

Almost increasing condition on $rac{(x,t)}{t^p}$

Extension of classical boundedness results to generalized Orlicz spaces

Abstract

We prove that the Hardy-Littlewood maximal operator is bounded in the weighted generalized Orlicz space if the weight satisfies the classical Muckenhoupt condition $A_p$ and $t \to \frac{\varphi(x,t)}{t^p}$ is almost increasing in addition to the standard conditions.