$L^p-L^q$ estimates for heat equation - regularizing effect 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.

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 :)
