Some remarks on the heat flow for functions and forms
by Anton Thalmaier

This note is concerned with the differentiation of heat semigroups on Riemannian manifolds. In particular, the relation dPtf = Ptdf 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. Even if M is flat and f smooth of compact support, the infinity norm of dPtf can in general not be estimated on compact time intervals in terms of f or df.

Electron. Comm. Probab.  3  (1998)  43-49

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