Nontangential limits and Fatou-type theorems on post-critically finite self-similar sets

TL;DR

This paper investigates boundary behavior of harmonic functions on self-similar fractal sets, establishing nontangential limits and Fatou theorems analogous to classical results in harmonic analysis.

Contribution

It extends classical boundary limit theorems to harmonic functions on p.c.f. self-similar fractals with a new proof of nontangential limits and Fatou theorems.

Findings

Proves existence of nontangential limits for Poisson integrals on p.c.f. sets.

Establishes Fatou-type theorems for bounded harmonic functions.

Demonstrates boundary behavior analogous to classical harmonic analysis results.

Abstract

In this paper we study the boundary limit properties of harmonic functions on $\mathbb R_+\times K$, the solutions $u(t,x)$ to the Poisson equation \[ \frac{\partial^2 u}{\partial t^2} + \Delta u = 0, \] where $K$ is a p.c.f. set and $\Delta$ its Laplacian given by a regular harmonic structure. In particular, we prove the existence of nontangential limits of the corresponding Poisson integrals, and the analogous results of the classical Fatou theorems for bounded and nontangentially bounded harmonic functions.