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Where can I find a proof of the following estimate $$\|S(t)v\|_{L^p(\Omega)}\leq C_{N,p,q} t^{-\frac{N}{2}\left(\frac{1}{q}-\frac{1}{p}\right)}\|v\|_{L^q(\Omega)}, $$ 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 generated by the heat equation with Dirichlet boundary condition ?

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This is standard, but the argument is short enough to fit in an answer. It is not restrictive to assume that $0\in\Omega$. Denote by $Q(t,x,y)$ the heat kernel associated with the Dirichlet Laplacian on $\Omega$ (this is simply the solution to the heat equation on $\Omega$ with Dirichlet b.c. and the delta function as initial data) and by $P(t,x,y)=ct^{-n/2}\exp(-|x-y|^2/4t)$ the heat kernel on $\mathbb{R}^n$, which is known by Fourier transform. By the maximum principle one obtains easily that $0\le Q\le P$ for all $t>0$ and $x,y\in\Omega$. The estimate follows immediately from this gaussian bound for $Q$.

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    $\begingroup$ Carro Piero, it is a bit provocative to call the kernels $p$ and $q$, when these letters denote the indices of the Lebesgue norms. $\endgroup$ Commented Mar 28, 2016 at 20:13
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    $\begingroup$ You are right, I noticed after I posted :). Let me change the names $\endgroup$ Commented Mar 28, 2016 at 20:20

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