Remarks on the Moser-Trudinger inequality

Abstract

We extend the Moser-Trudinger inequality (Equation Presented) to any Euclidean domain satisfying Poincaré's inequality (Equation Presented) 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 Ω: = ℝ x (-1, 1).