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

Abstract

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 $u\in C^{\infty }(M)$: $$\Vert \nabla u{\Vert }_p^p\le \Vert u{\Delta }_{p}u{\Vert }_1\in [0,\infty ],$$ 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 Schrödinger semigroups.

arXiv commenttitle changed, fixed typos, abstract and introduction rewritten, exposition improved according to referees suggestions, added Proposition 4 on Kato-Simon inequality on complete manifolds

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