Piecewise harmonic subharmonic functions and positive Cauchy transforms

Julius Borcea and Rikard Bøgvad

Vol. 240 (2009), No. 2, 231–265

DOI: 10.2140/pjm.2009.240.231

Abstract

We give a local characterization of the class of functions having positive distributional derivative with respect to z that are almost everywhere equal to one of finitely many analytic functions and satisfy some mild nondegeneracy assumptions. As a consequence, we give conditions that guarantee that any subharmonic piecewise harmonic function coincides locally with the maximum of finitely many harmonic functions and we describe the topology of their level curves. These results are valid in a quite general setting as they assume no à priori conditions on the differentiable structure of the support of the associated Riesz measures. We also discuss applications to positive Cauchy transforms and we consider several examples and related problems.

Keywords

subharmonic functions, piecewise analytic functions, positive Cauchy transforms

Mathematical Subject Classification 2000

Primary: 31A05

Secondary: 31A35, 30E20, 34M40

Milestones

Received: 30 November 2006

Revised: 3 February 2009

Accepted: 4 February 2009

Published: 4 March 2009