Aubin type almost sharp Moser-Trudinger inequality revisited

TL;DR

This paper presents a new proof of the almost sharp Moser-Trudinger inequality on compact Riemannian manifolds, extending the approach to higher order Sobolev spaces and manifolds with boundary, with relaxed smoothness conditions.

Contribution

It introduces a novel proof technique based on Euclidean sharp inequalities, allowing lower smoothness requirements and broader applicability to boundary conditions.

Findings

New proof of the almost sharp Moser-Trudinger inequality

Extension to higher order Sobolev spaces

Applicability to manifolds with boundary under various conditions

Abstract

We give a new proof of the almost sharp Moser-Trudinger inequality on compact Riemannian manifolds based on the sharp Moser inequality on Euclidean spaces. In particular we can lower the smoothness requirement of the metric and apply the same approach to higher order Sobolev spaces and manifolds with boundary under several boundary conditions.