Sharp Hardy-Littlewood-Sobolev inequalities on compact CR manifold

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 ${\mathcal{L}}_{\mathcal{\theta }}=b_n{\mathrm{\Delta }}_b+R$, ${\mathcal{Y}}_{\alpha }(M)$ is sharp constant, ${\mathrm{\Delta }}_b$ is Sublaplacian and $R$ is Tanaka-Webster scalar curvature. For the diagonal case $f=g$, we prove that $$\mathcal{Y}$\alpha(M)\geq $\mathcal{Y}$\alpha($\mathbb{S}$^{2n+1})$(theunitcomplexsphereof$$\mathbb{C}$^{n+1}$)and$$\mathcal{Y}$\alpha(M)$canbeattainedif$$\mathcal{Y}$\alpha(M)> $\mathcal{Y}$\alpha($\mathbb{S}$^{2n+1})$\mathrm{.}Particular,if$\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.