Consider the Dirichlet or Neumann Laplacian on a manifold with boundary. Suppose we have some estimate of the form $$||e^{t\Delta} f||_{L^p} \leq C(t)||f||_{L^q}$$ for some $p, q$. For a specific example, let $p = \infty, q = 2$. My question is, if and when we can use estimates like the above to calculate estimates like $$||\nabla e^{t\Delta} f||_{L^p} \leq C(t)||f||_{L^q}$$ for the same $p, q$.
PS: I have almost no background on probabilistic methods, so I would appreciate detailed references in that area (someone told me there are certain probabilistic methods of attack to these kinds of problems, that is why I have hesitatingly tagged probability as well). I am more or less okay with functional analysis/differential geometry/pde's.
Further edit: Internet search reveals that there is something called Bismut's formula which deals with something like this. Maybe, if that were translated in the semigroup language, it could lead to some kind of answer.
