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

Question

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.

Answer 1

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.