Extremals for singular Onofri type inequality in weighted Sobolev spaces

The $N$-dimensional Euclidean Onofri inequality was proved by Del Pino and Dolbeault for smooth compactly supported functions in ${\mathbb{R}}^N$ with $N\ge 2$. In this paper, we consider the singular version introduced by Lam and Lu, and extend it to a natural weighted Sobolev space. Next, we establish a sharp singular Carleson-Chang type inequality and prove its equivalence with the singular $N$-dimensional Euclidean Onofri inequality. Furthermore, we prove the existence of nontrivial, non-constant optimizers for the functional associated with the singular Euclidean Onofri inequality, answering open questions left by Del Pino and Dolbeault in the non-singular case, and by Lam and Lu in the case with singularities. To this end, we exploit the classification result obtained for a singular Liouville-type equation involving the $N$-Laplacian. We also prove the conformal invariance of the functional, establish the sharpness of the singular $N$-dimensional Euclidean Onofri inequality in terms of optimal constants, and state several consequences, including a sharp Poincaré inequality.