Hypoelliptic Bi-Invariant Laplacians on Infinite Dimensional Compact Groups

Bendikov, A. ; Saloff-Coste, L.

Canadian journal of mathematics, Tome 58 (2006) no. 4, pp. 691-725

Résumé

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.

Détail

DOI : 10.4153/CJM-2006-029-9

Mots-clés : 60B15, 43A77, 35H10, 46F25, 60J45, 60J6

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