Boundary values of holomorphic distributions in negative Lipschitz classes

TL;DR

This paper investigates the boundary behavior of holomorphic distributions in negative Lipschitz classes, linking it to Wiener series and Hausdorff contents, with implications for understanding boundary limits of such functions.

Contribution

It introduces new boundary value criteria for holomorphic distributions in negative Lipschitz classes using Wiener series and Hausdorff content analysis.

Findings

Characterization of boundary behavior of holomorphic distributions

Connection between Wiener series and boundary limits

Development of a new partition of unity technique

Abstract

We consider the behaviour at a boundary point of an open subset $U\subset\mathbb{C}$ of distributions that are holomorphic on $U$ and belong to what are called negative Lipschitz classes. The result explains the significance for holomorphic functions of series of Wiener type involving Hausdorff contents of dimension between $0$ and $1$. We begin with a survey about function spaces and capacities that sets the problem in context and reviews the relevant general theory. The techniques used include the construction of a special partition of the identity that may be of independent interest.