Heat equations and the Weighted $\bar\partial$-problem

Andrew Raich

doi:10.3934/cpaa.2012.11.885

Article Contents

2012, Volume 11, Issue 3: 885-909. Doi: 10.3934/cpaa.2012.11.885

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Heat equations and the Weighted $\bar\partial$-problem

Andrew Raich^1,

1.
Department of Mathematical Sciences, University of Arkansas, SCEN 301, Fayetteville, AR 72701

Received: May 31, 2010

Revised: August 31, 2011

Published: April 2012

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Abstract

The purpose of this article is to establish regularity and pointwise upper bounds for the (relative) fundamental solution of the heat equation associated to the weighted $\bar\partial$-operator in $L^2(C^n)$ for a certain class of weights. The weights depend on a parameter, and we find pointwise bounds for heat kernel, as well as its derivatives in time, space, and the parameter. We also prove cancellation conditions for the heat semigroup. We reduce the $n$-dimensional case to the one-dimensional case, and the estimates in one-dimensional case are achieved by Duhamel's principle and commutator properties of the operators. As an application, we recover estimates of the □$_b$-heat kernel on polynomial models in $C^2$.

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Mathematics Subject Classification: Primary: 32W30; Secondary: 32W05, 35K15.

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