The Dirichlet heat semigroup, $L^1_\delta$, and the dimension shift phenomenon

Question

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.

Answer 1

It seems dimensional analysis already reveals the exponent behaviour. If we use $m$ (say) to denote the unit of length, then an unweighted $L^p$ norm has units $m^{n/p}$, while a weighted $L^p$ norm has units $m^{(n+1)/p}$. The Laplacian $A$ has units $m^{-2}$, so time should have units $m^2$ in order for the exponent in $e^{tA}$ to be dimensionless. This soon predicts the right exponents for both estimates.

A bit more directly; a ball of radius $r$ has unweighted volume comparable to $r^n$, but has weighted volume comparable to $r^{n+1}$ if it is near the boundary (and the boundary is where all the "action" takes place). This already largely explains the dimension shifting phenomenon (noting also that at time t, a heat flow will have spread things out at the spatial scale of $r \sim t^{1/2}$.)

One can make the above dimensional analysis arguments rigorous by a scaling argument, at least in the model case when U is a half-space. These arguments do not actually prove the above estimates, but they show that the specified powers of $t$ are the only possible choices for such an estimate to be true.