On the concentration-compactness principle for Folland-Stein spaces and for fractional horizontal Sobolev spaces

In this paper we establish some variants of the celebrated concentration--compactness principle of Lions - CC principle briefly - in the classical and fractional Folland-Stein spaces. In the first part of the paper, following the main ideas of the pioneering papers of Lions, we prove the CC principle and its variant, that is the CC principle at infinity of Chabrowski, in the classical Folland-Stein space, involving the Hardy-Sobolev embedding in the Heisenberg setting. In the second part, we extend the method to the fractional Folland-Stein space. The results proved here will be exploited in a forthcoming paper to obtain existence of solutions for local and nonlocal subelliptic equations in the Heisenberg group, involving critical nonlinearities and Hardy terms. Indeed, in this type of problems a triple loss of compactness occurs and the issue of finding solutions is deeply connected to the concentration phenomena taking place when considering sequences of approximated solutions.