Moser-Trudinger inequalities: from local to global

TL;DR

This paper explores how local Moser-Trudinger inequalities on Riemannian manifolds can be extended to global inequalities, with applications to Hadamard manifolds and potential in sub-Riemannian contexts like the Heisenberg group.

Contribution

It introduces the concept of local Moser-Trudinger inequalities on manifolds and provides conditions for extending them globally, including applications to Hadamard and sub-Riemannian manifolds.

Findings

Extension of local to global inequalities under Poincaré or stronger norm assumptions

Application to Hadamard manifolds demonstrating the extension

Potential applicability to sub-Riemannian geometries like the Heisenberg group

Abstract

Given a general complete Riemannian manifold $M$, we introduce the concept of "local Moser-Trudinger inequality on $W^{1,n}(M)$". We show how the validity of the Moser-Trudinger inequality can be extended from a local to a global scale under additional assumptions: either by assuming the validity of the Poincar\'e inequality, or by imposing a stronger norm condition. We apply these results to Hadamard manifolds. The technique is general enough to be applicable also in sub-Riemannian settings, such as the Heisenberg group.