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Hypoelliptic Bi-Invariant Laplacians on Infinite Dimensional Compact Groups

Published online by Cambridge University Press:  20 November 2018

A. Bendikov  and
L. Saloff-Coste
A. Bendikov
Affiliation:
Instytut Matematyczny, UniwersytetuWrocławskiego, Poland e-mail: bendikov@math.uni.wroc.pl
L. Saloff-Coste
Affiliation:
Department of Mathematics, Cornell University, Ithaca, NY 14853-4201 U.S.A. e-mail: lsc@math.cornell.edu
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Abstract

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On a compact connected group $G$, consider the infinitesimal generator $-L$ of a central symmetric Gaussian convolution semigroup ${{\left( {{\mu }_{t}} \right)}_{t>0}}$. Using appropriate notions of distribution and smooth function spaces, we prove that $L$ is hypoelliptic if and only if ${{\left( {{\mu }_{t}} \right)}_{t>0}}$ is absolutely continuous with respect to Haar measure and admits a continuous density $x\mapsto {{\mu }_{t}}\left( x \right),t>0$, such that ${{\lim }_{t\to 0}}t\log {{\mu }_{t}}\left( e \right)=0$. In particular, this condition holds if and only if any Borel measure $u$ which is solution of $Lu=0$ in an open set $\Omega $ can be represented by a continuous function in $\Omega $. Examples are discussed.

Keywords

60B1543A7735H1046F2560J4560J6
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 58 , Issue 4 , 01 August 2006 , pp. 691 - 725
Copyright
Copyright © Canadian Mathematical Society 2006

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