How to find a measure from its potential

TL;DR

This paper extends the mathematical framework for reconstructing measures from their logarithmic potentials, especially on rectifiable curves, by generalizing Laplacian representations and singular integral equations to less smooth curves.

Contribution

It generalizes the representation of the generalized Laplacian for measures supported on any rectifiable curve and extends the theory of singular integral equations to Ahlfors regular curves.

Findings

Representation of the generalized Laplacian on rectifiable curves

Extension of singular integral equation theory to Ahlfors regular curves

Characterization of bounded solutions on arcs

Abstract

We consider the problem of finding a measure from the given values of its logarithmic potential on the support. It is well known that a solution to this problem is given by the generalized Laplacian. The case of our main interest is when the support is contained in a rectifiable curve, and the measure is absolutely continuous with respect to the arclength on this curve. Then the generalized Laplacian is expressed by a sum of normal derivatives of the potential. Such representation was available for smooth curves, and we show it holds for any rectifiable curve in the plane. We also relax the assumptions imposed on the potential. Finding a measure from its potential often leads to another closely related problem of solving a singular integral equation with Cauchy kernel. The theory of such equations is well developed for smooth curves. We generalize this theory to the class of Ahlfors regular curves and arcs, and characterize the bounded solutions on arcs.