Integral equations on compact CR manifolds

Assume that $ M $ is a CR compact manifold without boundary and CR Yamabe invariant $ \mathcal{Y}(M) $ is positive. Here, we devote to study a class of sharp Hardy-Littlewood-Sobolev inequality as follows

$ \begin{equation*} \Bigl| \int_M\int_M [G_\xi^\theta(\eta)]^{\frac{Q-\alpha}{Q-2}} f(\xi) g(\eta) dV_\theta(\xi) dV_\theta(\eta) \Bigr| \leq \mathcal{Y}_\alpha(M) \|f\|_{L^{\frac{2Q}{Q+\alpha}}(M)} \|g\|_{L^{\frac{2Q}{Q+\alpha}}(M)}, \end{equation*} $

where $ G_\xi^\theta(\eta) $ is the Green function of CR conformal Laplacian $ \mathcal{L_\theta} = b_n\Delta_b+R $, $ \mathcal{Y}_\alpha(M) $ is sharp constant, $ \Delta_b $ is Sublaplacian and $ R $ is Tanaka-Webster scalar curvature. For the diagonal case $ f = g $, we prove that $ \mathcal{Y}_\alpha(M)\geq \mathcal{Y}_\alpha(\mathbb{S}^{2n+1}) $ (the unit complex sphere of $ \mathbb{C}^{n+1} $) and $ \mathcal{Y}_\alpha(M) $ can be attained if $ \mathcal{Y}_\alpha(M)> \mathcal{Y}_\alpha(\mathbb{S}^{2n+1}) $. So, we got the existence of the Euler-Lagrange equations

$ \begin{equation} \varphi^{\frac{Q-\alpha}{Q+\alpha}}(\xi) = \int_M [G_\xi^\theta(\eta)]^{\frac{Q-\alpha}{Q-2}}\varphi(\eta)\ dV_\theta, \quad 0<\alpha<Q. ~~~(1) \end{equation} $

Moreover, we prove that the solution of (1) is $ \Gamma^\alpha(M) $. Particular, if $ \alpha = 2 $, the previous extremal problem is closely related to the CR Yamabe problem. Hence, we can study the CR Yamabe problem by integral equations.