HW. Due October 25

Exercise 1. Let $L$ be a locally subelliptic and essentially self-adjoint diffusion operator. Let $P_t$ be the semigroup generated by $L$ . By using the maximum principle for parabolic pdes, prove that if $f \ge 0$ is in $L^2$ , then $P_t f \ge 0$ .

Exercise 2: Let $L$ be an essentially self-adjoint diffusion operator. Denote by $P_t^{(p)}$ the semigroup generated by $L$ in $L^p$ .

Show that for each $f \in \mathbf{L}_{\mu}^{1}$ , the $\mathbf{L}_{\mu}^{1}$ -valued map $t \rightarrow \mathbf{P}^{(1)}_t f$ is continuous.
Show that for each $f \in \mathbf{L}_{\mu}^{p}$ , $1 < p < 2$ , the $\mathbf{L}_{\mu}^{p}$ -valued map $t \rightarrow \mathbf{P}^{(p)}_t f$ is continuous.
Finally, by using the reflexivity of $\mathbf{L}_{\mu}^{p}$ , show that for each $f \in \mathbf{L}_{\mu}^{p}$ and every $p \ge 1$ , the $\mathbf{L}_{\mu}^{p}$ -valued map $t \rightarrow \mathbf{P}^{(p)}_t f$ is continuous.