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 R2. Moreover we extend the analysis of Adimurthi (2004) and Csató and Roy (2015a) considering Adimurthi-Druet type functionals on compact surfaces with conical singularities and discussing the existence of extremals for such functionals.