  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.
