Aubin Type Almost sharp Moser–Trudinger Inequality Revisited

Abstract

We give a new proof of the almost sharp Moser–Trudinger inequality on smooth compact Riemannian manifolds based on the sharp Moser inequality on Euclidean spaces and generalize it to Riemannian manifolds with continuous metrics and to higher order Sobolev spaces on manifolds with boundary under several boundary conditions. These generalizations can be applied to fourth order Q curvature equations in dimension 4.