A new way to express boundary values in terms of holomorphic functions on planar Lipschitz domains

TL;DR

This paper introduces a novel decomposition of boundary functions on Lipschitz domains into sums of holomorphic functions, extending classical results to the $L^p$ setting and providing new insights into the Cauchy transform and Dirichlet problem.

Contribution

It extends boundary function decomposition to Lipschitz domains in the $L^p$ setting and offers a new proof of a regularity theorem for Hardy spaces, with applications to the Cauchy transform.

Findings

Decomposition of boundary functions into holomorphic components on Lipschitz domains.

New proof of regularity theorem for Hardy spaces in higher dimensions.

Characterization of the kernel of the Cauchy transform on $L^p$ boundary functions.

Abstract

We decompose $p$ - integrable functions on the boundary of a simply connected Lipschitz domain $\Omega \subset \mathbb C$ into the sum of the boundary values of two, uniquely determined holomorphic functions, where one is holomorphic in $\Omega$ while the other is holomorphic in $\mathbb C \setminus \overline{\Omega}$ and vanishes at infinity. This decomposition has been described previously for smooth functions on the boundary of a smooth domain. Uniqueness of the decomposition is elementary in the smooth case, but extending it to the $L^p$ setting relies upon a classical albeit little-known regularity theorem for the holomorphic Hardy space $h^p(b\Omega)$ of planar domains for which we provide a new proof that is valid also in higher dimensions. An immediate consequence of our result will be a new characterization of the kernel of the Cauchy transform acting on $L^p(b\Omega)$. These results give a new perspective on the classical Dirichlet problem for harmonic functions and the Poisson formula even in the case of the disc. Further applications are presented along with directions for future work.