Boundedness of Wolff-type potentials and applications to PDEs

TL;DR

This paper establishes a sharp estimate for a generalized Wolff-type potential, providing a reduction principle that characterizes boundedness in rearrangement invariant spaces, with applications to the regularity of solutions to quasilinear elliptic PDEs.

Contribution

It offers a new, simplified proof of a key estimate and introduces a reduction principle linking potential boundedness to Hardy-type inequalities, impacting PDE regularity analysis.

Findings

Sharp rearrangement estimate for generalized Wolff-type potential

Reduction principle characterizing boundedness in rearrangement invariant spaces

Applications to local regularity of solutions to quasilinear elliptic PDEs

Abstract

We provide a short proof of a sharp rearrangement estimate for a generalized version of a potential of Wolff--Havin--Maz'ya type. As a consequence, we prove a reduction principle for that integral operators, that is, a characterization of those rearrangement invariant spaces between which the potentials are bounded via a one-dimensional inequality of Hardy-type. Since the special case of the mentioned potential is known to control precisely very weak solutions to a broad class of quasilinear elliptic PDEs of non-standard growth, we infer the local regularity properties of the solutions in rearrangement invariant spaces for prescribed classes of data.