L^(p)-interpolation inequalities and global Sobolev regularity results (with an appendix by Ognjen Milatovic)

On any complete Riemannian manifold $M$ and for all $p\in [2,\infty)$, we prove a family of second order $L^{p}$-interpolation inequalities that arise from the following simple $L^{p}$-estimate valid for every $u \in C^{\infty}(M)$: $$ \|\nabla u\|_{p}^p \leq \|u \Delta_{p} u\|_1\in [0,\infty], $$ where $\Delta_p$ denotes the $p$-Laplace operator. We show that these inequalities, in combination with abstract functional analytic arguments, allow to establish new global Sobolev regularity results for $L^p$-solutions of the Poisson equation for all $p\in (1,\infty)$, and new global Sobolev regularity results for the singular magnetic Schr\"odinger semigroups.