Sharp Hardy-Littlewood-Sobolev inequalities on compact CR manifold

TL;DR

This paper establishes sharp Hardy-Littlewood-Sobolev inequalities on compact CR manifolds, relates extremal constants to the CR Yamabe problem, and provides conditions for attainability of extremals, extending classical inequalities to CR geometry.

Contribution

The paper derives sharp Hardy-Littlewood-Sobolev inequalities on CR manifolds, identifies extremal constants, and links the case b1=2 to the CR Yamabe problem, offering new integral equation approaches.

Findings

quality constants are minimized on the sphere.

Extremal functions exist when the constant exceeds the sphere's value.

The case b1=2 relates to the CR Yamabe problem.

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_\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 $\mathcal{Y}_\alpha(M)> \mathcal{Y}_\alpha(\mathbb{S}^{2n+1})$. 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.