CR eigenvalue estimate and Kohn-Rossi cohomology II

Let X be a weakly pseudoconvex, compact, and connected CR manifold with a transversal CR \(S^1\) $S^1$ -action of real dimension \(2n-1\) $2n-1$ , where \(n\ge 2\) $n\ge 2$ . The Fourier components of the Kohn-Rossi cohomology, with respect to the \(S^1\) $S^1$ -action, introduced by Hsiao-Li [6], are closely related to the embedding problem of CR manifolds. In this paper, we continue our previous study [14] and provide a sharp estimate for the asymptotic growth order, denoted as \(O(m^q)\) $O(m^q)$ , of the dimension of the m-th Fourier components \(H^{0,q}_{b,m}(X)\) $H_{b,m}^{0,q}(X)$ of the Kohn-Rossi cohomology \(H^{0,q}_{b}(X)\) $H_b^{0,q}(X)$ as \(m\rightarrow +\infty \) $m\to +\infty$ . Together with our previous work [14], we present a comprehensive complete and sharp estimate for the growth order of the Fourier components \(H^{0,q}_{b,m}(X)\) $H_{b,m}^{0,q}(X)$ and \(H^{n-1,q}_{b,m}(X)\) $H_{b,m}^{n-1,q}(X)$ of the Kohn-Rossi cohomology \(H^{0,q}_{b}(X)\) $H_b^{0,q}(X)$ and \(H^{n-1,q}_b(X)\) $H_b^{n-1,q}(X)$ as \(m\rightarrow \infty \) $m\to \infty$ . Additionally, we derive a Serre-type duality theorem for \(S^1\) $S^1$ -equivariant CR vector bundles.