Nontangential Limits and Fatou-Type Theorems on Post-Critically Finite Self-Similar Sets

In this paper we study the boundary limit properties of harmonic functions on ℝ + × K , the solutions u ( t , x ) to the Poisson equation where K is a p.c.f. set and Δ its Laplacian given by a regular harmonic structure. In particular, we prove the existence of nontangential limits of the correspond...

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Bibliographic Details
Published inThe Journal of fourier analysis and applications Vol. 18; no. 2; pp. 240 - 265
Main Author Sáenz, Ricardo A.
Format Journal Article
LanguageEnglish
Published Boston SP Birkhäuser Verlag Boston 01.04.2012
Subjects
Abstract Harmonic Analysis
Approximations and Expansions
Fourier Analysis
Mathematical Methods in Physics
Mathematics
Mathematics and Statistics
Partial Differential Equations
Signal,Image and Speech Processing
28A80
31B2
Fractals
P.c.f. sets
Poisson integrals
Boundary behavior of harmonic functions
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Summary:In this paper we study the boundary limit properties of harmonic functions on ℝ + × K , the solutions u ( t , x ) to the Poisson equation where K is a p.c.f. set and Δ 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.
ISSN:1069-5869
1531-5851
DOI:10.1007/s00041-011-9194-1