## 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.
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