On $L^p$-uniqueness of symmetric diffusion operators on Riemannian manifolds

Let $M$ be a complete Riemannian manifold of dimension $d>1$, let $\mu$ be a measure on $M$ with density $\exp U$ with respect to the Riemannian volume, and let $\mathscr Lf=\Delta f+\langle b,\nabla f\rangle$, where $U\in H^{p,1}_{\mathrm{loc}}(M)$ and $b=\nabla U$. It is shown that in the case $p>d$ and $q\in[p',p]$ the operator $\mathscr L$ on the domain $C_0^\infty(M)$ has a unique extension generating a $C_0$-semigroup on $L^q(M,\mu)$, that is, the set $(\mathscr L-I)(C_0^\infty(M))$ is dense in $L^q(M,\mu)$. In particular, the operator $\mathscr L$ is essentially self-adjoint on $L^2(M,\mu)$. A similar result is proved for elliptic operators with non-constant second order part that are formally symmetric with respect to some measure.