# Heat kernel estimates and Gaussian estimates for semigroups, good reference?

Hi, it seems like a big field and I'm having trouble getting some solid/classic references to get me started.

If $U \subset \mathbb{R}^d$ is a bounded domain with, say, $C^2$-boundary $\partial U$ and $(S(t),t \ge 0)$ is the Dirichlet heat semigroup on $L^p(U)$ then $(S(t) f)(x) = \int_U G_U(t,x,y) f(y)\,dy$ for $f \in L^p(U)$ where $G_U$ is the Dirichlet heat kernel.

I would like to find a good reference for bounds of the type: $$\left|\frac{\partial G_U}{\partial \nu_y} (t,x,y)\right| \le C_1 t^{-(d + k)/2} \exp\left(- \frac{|x-y|^2}{C_2 t}\right)$$ It seems like it should be classic result? I found Aronson's 1968 paper but it only contains estimates for the kernel and not the 'derivatives'. I can find lots of recent papers on manifolds and such but I am just looking for a solid and accessible reference for my simple case.

Further, if one has a semigroup $(T(t), t \ge 0)$ with a kernel $k(t,x,y)$ that has a pointwise Gaussian estimate $|k(t,x,y)| \le c_1 e^{\omega t} t^{-d/2} e^{-|x-y|^2/(c_2 t)}$, do the estimates for the 'derivatives' as above follow readily? or are more assumptions needed? Again, a reference to point me in the right direction would be greatly appreciated.

Thanks.

I would like to know a complete answer to this question myself, since this is useful in a number of situations. From what I know, contrary to estimates for the kernel which are a quite general phenomenon (see the classical book by Davies for your case), estimates for the derivatives are much more subtle and are connected with several properties of the generator such as $$L^p$$ boundedness for the associated Riesz operator when $$p>2$$ (while when $$p<2$$ the estimates for the derivatives are not needed).