Subharmonic Configurations and Algebraic Cauchy Transforms of Probability Measures

TL;DR

This paper investigates subharmonic functions with Laplacians supported on null sets, showing they are locally piecewise harmonic and describing their support when derivatives satisfy algebraic equations, with implications for algebraic Cauchy transforms.

Contribution

It establishes conditions under which subharmonic functions with specific derivative properties are piecewise harmonic and characterizes their support in algebraic cases.

Findings

Subharmonic functions with certain derivatives are locally piecewise harmonic.

Such functions often coincide with the maximum of finitely many harmonic functions.

The support of these functions can be described explicitly when derivatives satisfy algebraic equations.

Abstract

We study subharmonic functions whose Laplacian is supported on a null set and in connected components of of the complement to the support admit harmonic extensions to larger sets. We prove that if such a function has a piecewise holomorphic derivative, then it is locally piecewise harmonic. In generic cases it coincides locally with the maximum of finitely many harmonic functions. Moreover, we describe the support when the holomorphic derivative satisfies a global algebraic equation. The proofs follow classical patterns and our methods may also be of independent interest.