**Exercise 1. ***Let be a locally subelliptic and essentially self-adjoint diffusion operator. Let be the semigroup generated by . By using the maximum principle for parabolic pdes, prove that if is in , then .*

**Exercise 2: ***Let be an essentially self-adjoint diffusion operator. Denote by the semigroup generated by in .*

*Show that for each , the -valued map is continuous.**Show that for each , , the -valued map is continuous.**Finally, by using the reflexivity of , show that for each and every , the -valued map is continuous.*