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

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 \[ \frac{\partial^2 u}{\partial t^2} + Δu = 0, \] 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.

arXiv comment22 pages

References in corpus (1)

• Analysis of the Laplacian and Spectral Operators on the Vicsek Set