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

## TL;DR

This paper establishes new $L^{p}$-interpolation inequalities on complete Riemannian manifolds, leading to improved global Sobolev regularity results for solutions to the Poisson equation and magnetic Schrödinger semigroups.

## Contribution

It introduces a family of second order $L^{p}$-interpolation inequalities derived from a simple $L^{p}$-estimate, enabling new regularity results on Riemannian manifolds.

## Key findings

- Proved $L^{p}$-interpolation inequalities for all $p \\in [2,\\infty)$.
- Established new global Sobolev regularity for $L^p$-solutions of the Poisson equation.
- Derived regularity results for magnetic Schrödinger semigroups.

## 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 $u \in C^{\infty}(M)$: $$ \|\nabla u\|_{p}^p \leq \|u \Delta_{p} u\|_1\in [0,\infty], $$ 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.