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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