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Hypoelliptic Bi-Invariant Laplacians on Infinite Dimensional Compact Groups
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- Journal:
- Canadian Journal of Mathematics / Volume 58 / Issue 4 / 01 August 2006
- Published online by Cambridge University Press:
- 20 November 2018, pp. 691-725
- Print publication:
- 01 August 2006
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- Article
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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.
