Pullback via flow as operator group

Question

Let $X$ be a vector field on a manifold $M$ that induces a complete flow $\Theta_t$. Then the operator family $\Theta_t^*$, $$\Theta_t^*u(x) = u(\Theta_t(x))$$ is a strongly continuous semigroup of operators on the space $C_0(M)$, and also on the spaces $L^p(M)$, $1 \leq p < \infty$ under some extra conditions (e.g. if $M$ is compact).

The infinitesimal generator is obviously the vector field $X$ as a differential operator of order $1$, on some domain that includes the smooth compactly supported functions.

But what is the domain of the infinitesimal generator?

Answer 1

The domain of definition is the set of $f\in L^p(M)$ such that $f$ is differentiable along almost each flow line of $X$ and $Xf$ (the Lie derivative) is again a function in $L^p(M)$. It should be the completion of the space of smooth compactly supported functions with respect to the (semi-)norm $\|Xf\|_{L^p}$, at least when the measure on $M$ is well behaved, for example if it is the density for a Riemannian metric of bounded geometry. Treat the kernel of this norm separately.