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.