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Brownian Motion and Harmonic Analysis on Sierpinski Carpets
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- Journal:
- Canadian Journal of Mathematics / Volume 51 / Issue 4 / 01 August 1999
- Published online by Cambridge University Press:
- 20 November 2018, pp. 673-744
- Print publication:
- 01 August 1999
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- Article
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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.
