Extremals in Hardy-Littlewood-Sobolev inequalities for stable processes.

Abstract: In this talk, we prove the existence of an extremal function in the Hardy-Littlewood-Sobolev inequality for the energy associated with an stable operator, via a concentration-compactness principle for stable processes. This result can be seen as the first step to study existence of positive solutions of the corresponding nonlocal equations with critical nonlinearities perturbed by lower order terms.