In this article we investigate some effects of the lack of compactness in the critical fractional Folland-Stein-Sobolev embedding in general domains in the Heisenberg group \(\displaystyle (\mathbb {H}^N,\circ )\) with homogeneous dimension \(\displaystyle Q=2N+2\) . We consider the fractional horizontal Sobolev space \(\displaystyle HW^{s,2}(\mathbb {H}^N)\) for \(\displaystyle 0<s<1\) . It is well known that, for \(\displaystyle Q>2s\) the embedding \(\displaystyle HW^{s,2}(\mathbb {H}^N)\hookrightarrow L^q(\mathbb {H}^N)\) is continuous for \(\displaystyle q\le 2_s^*\) , and for bounded domains \(\displaystyle \Omega \subset \mathbb {H}^N\) , the embedding \(\displaystyle HW_0^{s,2}(\Omega )\hookrightarrow L^q(\Omega )\) is compact for \(\displaystyle q<2_s^*\) , where \(\displaystyle 2_s^*=\frac{2Q}{Q-2s}\) is the fractional Folland-Stein-Sobolev critical exponent. However, the compactness of the embedding fails in the critical case \(\displaystyle q=2_s^*\) . Here we restore the compactness by using De Giorgi’s \(\displaystyle \Gamma \) -convergence techniques and some nonlocal tail estimates in \(\displaystyle \mathbb {H}^N.\)
