On the regularity of maximal operators

Abstract

We study the regularity of the bilinear maximal operator when applied to Sobolev functions, proving that it maps $W^{1,p}(\mathbb{R})\times W^{1,q}(\mathbb{R})\to W^{1,r}(\mathbb{R})$ with $1<p,q<\infty$ and $r\ge 1$, boundedly and continuously. The same result holds on ${\mathbb{R}}^n$ when $r>1$. We also investigate the almost everywhere and weak convergence under the action of the classical Hardy-Littlewood maximal operator, both in its global and local versions.

10 pages

References in corpus (1)

• On the regularity of maximal operators

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• Regularity of maximal operators: recent progress and some open problems
• Regularity and continuity of local Multilinear Maximal type operator
• Regularity and continuity of the multilinear strong maximal operators