The singular Moser- Trudger Inequality on simply connected domains

TL;DR

This paper simplifies and generalizes the proof of the singular Moser-Trudinger inequality on simply connected domains using complex analysis, making the results more accessible and extending previous work.

Contribution

The authors provide a simpler, more self-contained proof of the singular Moser-Trudinger inequality on simply connected domains, extending Flucher's approach.

Findings

Simplified proof using complex analysis

Generalization of previous results to simply connected domains

Enhanced accessibility of the singular Moser-Trudinger inequality

Abstract

In this paper the authors complete their study of the singular Moser-Trudinger embedding [G. Csato and P. Roy, Extremal functions for the singular Moser-Trudinger inequality in 2 dimensions, Calc. Var. Partial Differential Equations, DOI 10.1007/s00526-015-0867-5], abbreviated [CR]. The proof in [CR] is however far too technical and complicated for simply connected domains. Here we give a much simpler and more self-contained proof using complex analysis, which also generalizes the corresponding proof given by Flucher for such domains. This should make [CR] more easily accessible.