Inserted: 12 nov 2021
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.