Some remarks on the heat flow for functions and forms (Q1269721)

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, \(\| dP_tf\|_\infty\) cannot be estimated on compact time intervals in terms of \(f\) or \(df\).