Maximal potentials, maximal singular integrals, and the spherical maximal function

TL;DR

This paper introduces maximal potentials and proves their boundedness from L^p to Sobolev spaces, then applies these results to analyze the boundedness of the spherical maximal operator in Sobolev spaces.

Contribution

The paper defines maximal potentials and establishes their boundedness, providing new tools to study the spherical maximal operator in Sobolev spaces.

Findings

Maximal potentials are bounded from L^p to ot{W}^{1,p} for specific p ranges.

Application of maximal potentials yields boundedness results for the spherical maximal operator.

Provides a new approach to analyze maximal functions in Sobolev spaces.

Abstract

We introduce a notion of maximal potentials and we prove that they form bounded operators from $L^p$ to the homogeneous Sobolev space $\dot{W}^{1,p}$ for all $n/(n-1)<p<n$. We apply this result to the problem of boundedness of the spherical maximal operator in Sobolev spaces.