Maximal operator in Musielak--Orlicz--Sobolev spaces

Abstract

We study the Hardy-Littlewood maximal operator in the Musielak-Orlicz-Sobolev space $W^{1,\varphi}(\mathbb{R}^n)$. Under some natural assumptions on $\varphi$ we show that the maximal function is bounded and continuous in $W^{1,\varphi}(\mathbb{R}^n)$.