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)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$W<^>{1,n}(M)$$\end{document}". 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 & eacute; 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.