I have a few questions concerning the heat semigroup $e^{\Delta t}$ on Morrey spaces.
(1) I read that the heat semigroup on $ M^{p}_{q}(\mathbb{R}^n):= \left\{ f \in L^{q}_{loc}(\mathbb{R}^n): \mbox{ } ||{f}||_{M^{p}_{q}(\mathbb{R}^n)}^q:= \sup_{x \in \mathbb{R}^n, 0<R \leq 1}{R^{n(q/p-1)} \int_{B(x,R)}{|f|^q}} < \infty \right\} $, where $1\leq q \leq p \leq \infty$, is not a strongly continuous semigroup. How does one prove this?
(2) Next define the space $D^{p}_{q}(\mathbb{R}^n)= \left\{ f \in M^{p}_{q}(\mathbb{R}^n): \mbox{ } || \tau_{y}f-f ||_{M^{p}_{q}(\mathbb{R}^n)} \rightarrow 0 \mbox{ as } | y |\rightarrow 0 \right\}$. I see that this space is the maximal closed subspace of $M^{p}_{q}(\mathbb{R}^n)$ on which the family of translations forms a $C^0$-semigroup. But I don't know how to proof the same result on the heat semigroup.
Do you have any ideas?
Thanks a lot!
The functions in Morrey space cannot be approximated by functions in C^{\infty}, hence the heat semigroup does not a C_0-semigroup in Morrey spaces.