Extremal functions for singular Moser-Trudinger embeddings

We study Moser-Trudinger type functionals in the presence of singular potentials. In particular we propose a proof of a singular Carleson-Chang type estimate by means of Onofri’s inequality for the unit disk in $\mathbb{R}^2$. Moreover we extend the analysis of [1] and [8] considering Adimurthi-Druet type functionals on compact surfaces with conical singularities and discussing the existence of extremals for such functionals.