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

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