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Brownian Motion and Harmonic Analysis on Sierpinski Carpets

Published online by Cambridge University Press:  20 November 2018

Martin T. Barlow
Affiliation:
Department of Mathematics, University of British Columbia, Vancouver, BC, V6T 1Z2
Richard F. Bass
Affiliation:
Department of Mathematics, University of Connecticut, Storrs, CT 06269, USA
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Abstract

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We consider a class of fractal subsets of ${{\mathbb{R}}^{d}}$ formed in a manner analogous to the construction of the Sierpinski carpet. We prove a uniform Harnack inequality for positive harmonic functions; study the heat equation, and obtain upper and lower bounds on the heat kernel which are, up to constants, the best possible; construct a locally isotropic diffusion $X$ and determine its basic properties; and extend some classical Sobolev and Poincaré inequalities to this setting.

Type
Research Article
Copyright
Copyright © Canadian Mathematical Society 1999

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