Sharp Hardy-Littlewood-Sobolev inequalities on compact CR manifold

Abstract

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_ξ^θ(η)]^{\frac{Q-α}{Q-2}} f(ξ) g(η) dV_θ(ξ) dV_θ(η) \Bigr| \leq \mathcal{Y}_α(M) \|f\|_{L^{\frac{2Q}{Q+α}}(M)} \|g\|_{L^{\frac{2Q}{Q+α}}(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 ${\mathcal{Y}}_{\alpha }(M)\ge {\mathcal{Y}}_{\alpha }({\mathbb{S}}^{2n+1})$ (the unit complex sphere of ${\mathbb{C}}^{n+1}$) and ${\mathcal{Y}}_{\alpha }(M)$ can be attained if ${\mathcal{Y}}_{\alpha }(M)>{\mathcal{Y}}_{\alpha }({\mathbb{S}}^{2n+1})$. 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.

References in corpus (2)

• Hardy-Littlewood-Sobolev inequalities on compact Riemannian manifolds and applications
• An integral type Brezis-Nirenberg problem on the Heisenberg group