Two Weight Estimates for the Single Layer Potential on Lipschitz Surfaces with Small Lipschitz Constant

TL;DR

This paper establishes two weight estimates for the single layer potential on Lipschitz surfaces with small Lipschitz constant, providing conditions for solvability, uniqueness, and isomorphism in weighted spaces, including explicit weight functions.

Contribution

It introduces new conditions for two weight estimates on Lipschitz surfaces with small Lipschitz constant, extending results to weighted Lebesgue and Sobolev spaces with explicit weight functions.

Findings

Conditions for solvability and uniqueness in weighted spaces.

Single layer potential acts as an isomorphism under certain weights.

Generalization to power exponential weights beyond hyperplane cases.

Abstract

This article considers two weight estimates for the single layer potential --- corresponding to the Laplace operator in $\mathbf{R}^{N+1}$ --- on Lipschitz surfaces with small Lipschitz constant. We present conditions on the weights to obtain solvability and uniqueness results in weighted Lebesgue spaces and weighted homogeneous Sobolev spaces, where the weights are assumed to be radial and doubling. In the case when the weights are additionally assumed to be differentiable almost everywhere, simplified conditions in terms of the logarithmic derivative are presented, and as an application, we prove that the operator corresponding to the single layer potential in question is an isomorphism between certain weighted spaces of the type mentioned above. Furthermore, we consider several explicit weight functions. In particular, we present results for power exponential weights which generalize known results for the case when the single layer potential is reduced to a Riesz potential, which is the case when the Lipschitz surface is given by a hyperplane.