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∈ [2 , ∞) , we prove a family of second-order L p -interpolation inequalities that arise from the following simple L p -estimate valid for every u∈ C ∞ (M) : ‖∇u‖pp≤‖uΔpu‖1∈[0,∞],where Δ 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∈ (1 , ∞) , and new global Sobolev regularity results for the singular magnetic Schrödinger semigroups.