Uniqueness theorems for Cauchy integrals

TL;DR

This paper investigates the properties of reflectionless measures in the complex plane, establishing conditions under which such measures must be trivial and exploring their geometric and analytical characteristics.

Contribution

It proves that reflectionless measures with summable Cauchy maximal functions are trivial, provides a sharp example, and offers a partial geometric description of these measures.

Findings

Reflectionless measures with summable maximal functions are trivial.

Constructed an example with maximal function in weak L^1.

Connected reflectionless measures to sets of finite perimeter.

Abstract

If $\mu$ is a finite complex measure in the complex plane $\C$ we denote by $C^\mu$ its Cauchy integral defined in the sense of principal value. The measure $\mu$ is called reflectionless if it is continuous (has no atoms) and $C^\mu=0$ at $\mu$-almost every point. We show that if $\mu$ is reflectionless and its Cauchy maximal function $C^\mu_*$ is summable with respect to $|\mu|$ then $\mu$ is trivial. An example of a reflectionless measure whose maximal function belongs to the "weak" $L^1$ is also constructed, proving that the above result is sharp in its scale. We also give a partial geometric description of the set of reflectionless measures on the line and discuss connections of our results with the notion of sets of finite perimeter in the sense of De Giorgi.