Boundedness of Wolff-type potentials and applications to PDEs

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.

Automated summary

This article provides a short proof of a sharp rearrangement estimate for a generalized version of a potential of Wolff-Havin-Maz'ya type. It characterizes the potentials that are bounded between rearrangement invariant spaces via a one-dimensional inequality of Hardy-type. By controlling very weak solutions to a broad class of quasilinear elliptic PDEs of non-standard growth, the special case of the mentioned potential infers the local regularity properties of solutions in rearrangement invariant spaces for prescribed classes of data.