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Gaussian Estimates for the Heat Kernel of the Weighted Laplacian and Fractal Measures

Published online by Cambridge University Press:  20 November 2018

Alberto G. Setti
Alberto G. Setti*
Affiliation:
Department of Mathematics, Cornell University, Ithaca, New York14853, U.S.A.
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Abstract

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Let 0 < wbe a smooth function on a complete Riemannian manifold Mn, and define L = — Δ — ▽ (log w) and Rw =Ric - w -1 Hess w.In this paper we show that if Rw ≥ —nK, (K ≥0), then the positive solutions of (L + ∂/∂t)u —0 satisfy a gradient estimate of the same form as that obtained by Li and Yau ([LY]) when Lis the Laplacian. This is used to obtain a parabolic Harnack inequality, which in turn, yields upper and lower Gaussian estimates for the heat kernel of L.The results obtained are applied to study the LPmapping properties of t→ e-tL μfor measures μ which are α-dimensional in a sense that generalises the local uniform α-dimensionality introduced by R. S. Strichartz ([St2], [St3]).

Keywords

Primary: 58G1158C35secondary: 35K0528A75
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 44 , Issue 5 , 01 October 1992 , pp. 1061 - 1078
Copyright
Copyright © Canadian Mathematical Society 1992

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