Inequalities for Eigenvalues of the Sub-Laplacian on Strictly Pseudoconvex CR Manifolds

The sub-Laplacian plays a key role in CR geometry. In this paper, we investigate eigenvalues of the sub-Laplacian on bounded domains of strictly pseudoconvex CR manifolds, strictly pseudoconvex CR manifolds submersed in Riemannian manifolds. We establish some Levitin–Parnovski-type inequalities and Cheng–Huang–Wei-type inequalities for their eigenvalues. As their applications, we derive some results for the standard CR sphere $\mathbb{S}^{2n+1}$ in $\mathbb{C}^{n+1}$, the Heisenberg group $\mathbb{H}^n$, a strictly pseudoconvex CR manifold submersed in a minimal submanifold in $\mathbb{R}^m$, domains of the standard sphere $\mathbb{S}^{2n}$ and the projective space $\mathbb{F}P^m$.