Hilbert transforms and the Cauchy integral in euclidean space

TL;DR

This paper extends harmonic conjugates and Hilbert transforms to higher dimensions using differential forms, establishing existence, uniqueness, and invertibility results under boundary conditions.

Contribution

It introduces a higher-dimensional framework for harmonic conjugates and Hilbert transforms, with new invertibility results for generalized layer potential operators.

Findings

Existence and uniqueness of conjugates under boundary conditions

Invertibility of generalized double layer potential operators

Boundedness of related Hilbert transforms

Abstract

We generalize the notion of harmonic conjugate functions and Hilbert transforms to higher dimensional euclidean spaces, in the setting of differential forms and the Hodge-Dirac system. These conjugate functions are in general far from being unique, but under suitable boundary conditions we prove existence and uniqueness of conjugates. The proof also yields invertibility results for a new class of generalized double layer potential operators on Lipschitz surfaces and boundedness of related Hilbert transforms.