Weak and strong limit values

TL;DR

This paper investigates boundary limits of holomorphic or harmonic functions without assuming boundary smoothness, challenging classical approaches that rely on smooth boundary properties and kernel estimates.

Contribution

It introduces a framework for analyzing boundary limits without smooth boundary assumptions, expanding the understanding of boundary behavior in complex analysis.

Findings

Classical boundary limit results depend on boundary smoothness.

New approach removes the need for smooth boundary assumptions.

Potential for broader application in irregular domains.

Abstract

The classical results about the boundary values of holomorphic or harmonic functions on a domain $D$ state that under additional integrability assumptions these functions have limits along specific sets approaching boundary. The proofs of these results are based on properties of smooth boundaries used to define the approach regions and on estimates of representing kernels along these regions. This paper attempts to look at the situation when no assumptions about the boundary smoothness are made and, consequently, no natural definitions of approach regions could be given.