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Where can I find a proof of the following estimate $$\|S(t)v\|_{L^p(\Omega)}\leq C t^{-\frac{N}{2}\left(\frac{1}{q}-\frac{1}{p}\right)}\|v\|_{L^q(\Omega)}, $$ where $1\leq p<q<+\infty$, $\Omega\subset \mathbb{R}^N$ is an open bounded set and $\{S(t)\}_{t\geq 0}$ is the semigroup generate by the heat equation with Dirichlet boundary condition.

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3 Answers 3

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Let $N_t:\mathbb{R}^N\to \mathbb{R}$, $t>0$, be the function defined by $$N_t(x)=(4\pi t)^{-N/2}e^{-|x|^2/4t}.$$ Since $$\int_{\mathbb{R}^N} e^{-a|x|^2}dx=\left(\frac{\pi}{a}\right)^{N/2},\tag{1}\label{1}$$ we can see that $N_t\in L^1(\mathbb{R}^N)$ and $\|N_t\|_{ L^1(\mathbb{R}^N)}=1$.

We know that $S(t)v=N_t\ast v$. From Young's Inequality, we have $$\|S(t)v\|_{ L^p(\Omega)}\leq \|N_t\ast v\|_{ L^p(\Omega)}\leq \|N_t\|_{ L^m(\Omega)}\|v\|_{ L^q(\Omega)},$$ where $1+\frac{1}{p}=\frac{1}{m}+\frac{1}{q}.$

Now, we just have to estimate $\|N_t\|_{ L^m(\Omega)}$. From \eqref{1}, we can see that $$\|N_t\|_{ L^m(\Omega)}=(4\pi t)^{-N/2}\left(\int_{\mathbb{R}^N} e^{-\frac{m}{4t}|x|^2}dx\right)^{1/m}=(4\pi t)^{-N/2}\left(\frac{\pi}{\frac{m}{4t}}\right)^{N/2m}=C_{m,N}t^{-\frac{N}{2}\left(1-\frac{1}{m}\right)}=C_{m,N}t^{-\frac{N}{2}\left(\frac{1}{q}-\frac{1}{p}\right)}.$$

Hence, we have the result.

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The answer by the OP does not directly address their question because it uses the explicit form of the heat kernel on $\mathbb{R}^N$, while the question is posed for an arbitrary (bounded) domain $\Omega$. Interestingly there is a nice argument to deduce the estimate on $\Omega$ from that on $\mathbb{R}^N$ (see Davies, Heat Kernels and Spectral Theory, Theorem 2.1.6):

Let $\Delta$ be the Laplacian on $L^2(\mathbb{R}^N)$, $\Delta_\Omega$ the Laplacian on $L^2(\Omega)$ with Dirichlet boundary conditions, and $\chi$ the characteristic function of $\mathbb{R}^N\setminus \Omega$. By standard results on semigroup convergence, $e^{t(\Delta-n\chi)}f\to e^{t\Delta_\Omega}f$ as $n\to \infty$.

It follows from the Kato-Trotter product formula that $$ 0\leq e^{t\Delta-n\chi}f\leq e^{t\Delta} f $$ for all positive $f\in L^2(\mathbb{R}^N)$. In the limit $N\to\infty$ it follows that $e^{t\Delta_\Omega}f\leq e^{t\Delta}f$ for all positive $f\in L^2(\Omega)$, which is easily seen to be equivalent to $$ |e^{t\Delta_\Omega}f|\leq e^{t\Delta}|f| $$ for all $f\in L^2(\Omega)$.

This shows that the estimate from the question for the semigroup $(e^{t\Delta})$ implies the same bound for $(e^{t\Delta_\Omega})$.

An alternative approach is to notice that the estimate from the question is equivalent to the Sobolev inequality $$ \|f\|_{2N/(N-2)}\leq C\|\nabla f\|_2, $$ which can obviously be localized.

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  • $\begingroup$ Very nice. I didn't see that. Thanks to clarify it. $\endgroup$ Commented Mar 29, 2020 at 14:06
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I found the reference Davies, Heat Kernels and Spectral Theory provided by upstairs to be very useful. Actually, using the pointwise estimates of the heat kernel for the Dirichlet (or neumann) Laplacian in chapter 3 of this book, together with the Young's inequality of convolution, one can prove the L^p-L^q type estimate for heat semi-group. It is a good exercise then :)

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