Moser-Trudinger inequalities: from local to global

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 Poincaré 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.

Published in Annali di Matematica Pura ed Applicata, July 30, 2024. In this version we added the reference [G2] (cited on page 10)