Remarks on the Moser-Trudinger inequality

TL;DR

This paper extends the Moser-Trudinger inequality to certain Euclidean domains, explores its limitations under conformal metrics, and investigates extremals in unbounded domains, providing new theoretical insights.

Contribution

It generalizes the Moser-Trudinger inequality to Euclidean domains with Poincaré's inequality and analyzes extremals in unbounded domains like the infinite strip.

Findings

Extension of the inequality to specific Euclidean domains

Counterexamples for conformal metrics on the unit ball

Existence of extremals in the infinite planar strip

Abstract

We extend the Moser-Trudinger inequality to any Euclidean domain satisfying Poincar\'e's inequality. We find out that the same equivalence does not hold in general for conformal metrics on the unit ball, showing counterexamples. We also study the existence of extremals for the Moser-Trudinger inequalities for unbounded domains, proving it for the infinite planar strip.