Some Remarks on the Heat Flow for Functions and Forms

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 $df$.