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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.

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2 Answers

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; go figure).

This paper by ter Elst et al. gives some info in the case of elliptic operators, and on manifold this is studied in a 2004 paper by Auscher, Coulhon, Duong and Hofmann (Riesz transform on manifolds and heat kernel regularity) which should be on arXiv.

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Warning: the following response is that of a "googlist" not of an expert.

Although not classical (as per your request), perhaps the following two offer reasonable pointers? If you find these to be unhelpful, please let me know. I also had a longer listing available, if I find it, I will update my answer.

  1. Heat kernel expansion: user's manual: arXiv link
  2. Several of the papers of Grigoryan, e.g., this one
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Thanks. I actually have already quickly browsed through the new book "Heat kernel and analysis on manifolds" by Alexander Grigoryan at my local library. I didn't really find what I was looking for. I can crudely get out the estimate myself for the heat kernel in the trivial half-plane case but didn't want to waste more time as I am sure there is probably a good reference I could use. Paper 2 looks nice, I will have a deeper look at it, maybe it's hidden there. –  Dale Roberts Nov 12 '10 at 10:49
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