$$L^{p}$$ 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 $$u \in C^{\infty }(M)$$ : $$\begin{aligned} \Vert \nabla u\Vert _{p}^p \le \Vert u \Delta _{p} u\Vert _1\in [0,\infty ], \end{aligned}$$ 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 Schrodinger semigroups.