On the regularity of maximal operators

TL;DR

This paper investigates the regularity properties of the bilinear maximal operator on Sobolev spaces, establishing boundedness and continuity results, and explores convergence behaviors of the classical Hardy-Littlewood maximal operator.

Contribution

It proves the boundedness and continuity of the bilinear maximal operator on Sobolev spaces and analyzes convergence properties of the Hardy-Littlewood maximal operator.

Findings

Bilinear maximal operator maps Sobolev spaces to Sobolev spaces with boundedness.

Results extend to higher dimensions for r > 1.

Convergence properties of the Hardy-Littlewood maximal operator are characterized.

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 $r\geq 1$, 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.