show/hide this revision's text 5 Rollback to Revision 3 - Rolled back.

Added: Some additional background and motivation

I am somewhat curious what properties of $(e^{-tA})_{t \ge 0}$ change when considered in such weighted spaces. For example, things to check are: analyticity, ultracontractivity, Gaussian estimates, Hilbert-Schmidt property of $e^{-tA}$ in the $L^2$ case (follows from ultracontractivity), changes to the spectrum, etc. What else might be interesting to check?

The motivation is that one always sees the same examples in books on semigroup theory and heat kernels: Dirichlet heat semigroup and sometimes the Neumann heat semigroup (i.e. zero Neumann conditions built-in to $A$). I always find it instructive to see examples where a slight change of conditions can cause a subtle change in the properties.

For example, one thing that obviously changes is that the Dirichlet Laplacian $A$ on $L^2(U)$ is no longer self-adjoint on $L^2(U,\delta)$.

If there is something interesting that changes due to this dimension shift phenomenon, maybe it would be a nice additional example for someone's future book or lecture notes? :)

It seems one can deduce some of the properties from the papers by Krylov on the topic (although his notation is very overloaded) and of course, the papers and book by Quittner and Souplet.

I'm definitely going to have a go myself at this with Terry's great suggestion of a dimensional analysis approach :)
show/hide this revision's text 4 Added additional background.

Added: Some additional background and motivation

I am somewhat curious what properties of $(e^{-tA})_{t \ge 0}$ change when considered in such weighted spaces. For example, things to check are: analyticity, ultracontractivity, Gaussian estimates, Hilbert-Schmidt property of $e^{-tA}$ in the $L^2$ case (follows from ultracontractivity), changes to the spectrum, etc. What else might be interesting to check?

The motivation is that one always sees the same examples in books on semigroup theory and heat kernels: Dirichlet heat semigroup and sometimes the Neumann heat semigroup (i.e. zero Neumann conditions built-in to $A$). I always find it instructive to see examples where a slight change of conditions can cause a subtle change in the properties.

For example, one thing that obviously changes is that the Dirichlet Laplacian $A$ on $L^2(U)$ is no longer self-adjoint on $L^2(U,\delta)$.

If there is something interesting that changes due to this dimension shift phenomenon, maybe it would be a nice additional example for someone's future book or lecture notes? :)

It seems one can deduce some of the properties from the papers by Krylov on the topic (although his notation is very overloaded) and of course, the papers and book by Quittner and Souplet.

I'm definitely going to have a go myself at this with Terry's great suggestion of a dimensional analysis approach :)
show/hide this revision's text 3 Added assumption on p,q. Added clarification of zero b.c.

In relation to the question on the Hardy inequality and the answer by Terry Tao, I've always been curious about the following:

Let $U \subset \mathbb{R}^n$ be a bounded domain of class $C^2$, $(e^{-t A})_{t \ge 0}$ be the Dirichlet heat semigroup(s) on $L^p(U)$, $1 \le p \le \infty$. $A$ is the Dirichlet Laplacian (i.e. zero boundary conditions). Compare the following (where $\lesssim$ hides a constant dependent on $p,q,U$):

For $\varphi \in L^p(U)$ , and $1\le p \le q < \infty$, we have $$\| e^{-t A} \varphi\|_{q} \lesssim \|\varphi\|_p t^{-\frac{n}{2}(\frac{1}{p}-\frac{1}{q})}, \quad t >0.$$

and now on the weighted spaces $L^p(U,\delta)$ where $\delta(x):=\text{dist}(x,\partial U)$, $1\le p \le \infty$.

For $\varphi \in L^p_\delta(U)$ , and $1\le p \le q < \infty$, we have $$\| e^{-t A} \varphi\|_{q,\delta} \lesssim \|\varphi\|_{p,\delta} t^{-\frac{n+1}{2}(\frac{1}{p}-\frac{1}{q})}, \quad t >0.$$

Quittner and Souplet call this the dimension shift phenomenon: the weighted space estimates are similar to those in standard $L^p$-spaces in $n+1$ dimensions.

Question 1: Is there something subtle and interesting happening here?

It seems to be based on the following (sketched) observations: from the estimate$$|e^{-tA} \varphi(x)| \lesssim \|\phi\|_{\infty} \frac{\delta(x)}{\sqrt{t}},\quad x \in U, t > 0,\quad \varphi \in L^\infty(U),$$ and as $e^{-tA}$ is self-adjoint in $(L^2,(\cdot,\cdot))$,
$$\|e^{-tA} \varphi\|_1 = (e^{-tA} \varphi, \chi_{U}) = (\varphi, e^{-tA} \chi_U) \lesssim t^{-1/2}(\varphi,\delta)$$ so $$\|e^{-tA} \varphi\|_{\infty} = \|e^{-(t/2)A}(e^{(t/2)A)} \varphi)\|_\infty \lesssim t^{-n/2} \|e^{-(t/2)A} \varphi\|_1 \lesssim t^{-(n+1)/2}\|\varphi\|_{1,\delta}$$ and the weighted estimate is obtained by Holder's inequality and some additional rigour (see the book by Quittner/Souplet for details).

Question 2: The weight $\delta$ seems very special. What can be said in the case $\delta^\alpha$ where $\alpha > 1$? It seems a different argument is needed.

I would love to hear any insightful or interesting remarks about the above. Thanks.
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