Moser–Trudinger inequalities: from local to global

Given a general complete Riemannian manifold M, we introduce the concept of “local Moser–Trudinger inequality on W1,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.