Moser-Trudinger inequality on conformal discs

TL;DR

This paper characterizes when the Moser-Trudinger inequality holds on conformal discs and unbounded Euclidean subsets, linking it to bounds related to hyperbolic metrics.

Contribution

It provides a precise criterion for the validity of the Moser-Trudinger inequality in conformal discs based on hyperbolic metric bounds.

Findings

The inequality holds iff the metric is bounded above by the hyperbolic metric.

A necessary and sufficient condition is established for unbounded Euclidean subsets.

The results connect geometric bounds with functional inequalities.

Abstract

We show that the Moser-Trudinger inequality holds in a conformal disc if and only if the metric is bounded from above by the Hyperbolic metric. We also find a necessary and sufficient condition for the Moser-Trudinger inequality to hold in an unbounded subset of the two dimensional Euclidean space.