Some Remarks on the Heat Flow for Functions and Forms

Anton Thalmaier (University of Bonn)

Abstract


This note is concerned with the differentiation of heat semigroups on Riemannian manifolds. In particular, the relation $dP_tf=P_tdf$ is investigated for the semigroup generated by the Laplacian with Dirichlet boundary conditions. By means of elementary martingale arguments it is shown that well-known properties which hold on complete Riemannian manifolds fail if the manifold is only BM-complete. In general, even if $M$ is flat and $f$ smooth of compact support, $\Vert dP_tf\Vert_\infty$ cannot be estimated on compact time intervals in terms of $f$ or $df$.

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Pages: 43-49

Publication Date: July 18, 1998

DOI: 10.1214/ECP.v3-992

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