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

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

2010-11-10

arxiv.org/abs/1011.2474v3

J Fourier Anal Appl (2012) 18: 240-265

Mathematics

Classical Analysis and ODEs

22 pages

Scientific paper

10.1007/s00041-011-9194-1

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.

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