Let $X$ be a compact connected CR manifold with a transversal CR $S^1$-action of dimension $2n-1$, which is only assumed to be weakly pseudoconvex. Let $Box_b$ be the $overline{partial}_b$-Laplacian. Eigenvalue estimate of $Box_b$ is a fundamental issue both in CR geometry and analysis. In this paper, we are able to obtain a sharp estimate of the number of eigenvalues smaller than or equal to $lambda$ of $Box_b$ acting on the $m$-th Fourier components of smooth $(n-1,q)$-forms on $X$, where $min mathbb{Z}_+$ and $q=0,1,cdots, n-1$. Here the sharp means the growth order with respect to $m$ is sharp. In particular, when $lambda=0$, we obtain the asymptotic estimate of the growth for $m$-th Fourier components $H^{n-1,q}_{b,m}(X)$ of $H^{n-1,q}_b(X)$ as $m rightarrow +infty$. Furthermore, we establish a Serre type duality theorem for Fourier components of Kohn-Rossi cohomology which is of independent interest. As a byproduct, the asymptotic growth of the dimensions of the Fourier components $H^{0,q}_{b,-m}(X)$ for $ min mathbb{Z}_+$ is established. Compared with previous results in this field, the estimate for $lambda=0$ already improves very much the corresponding estimate of Hsiao and Li . We also give appilcations of our main results, including Morse type inequalities, asymptotic Riemann-Roch type theorem, Grauert-Riemenscheider type criterion, and an orbifold version of our main results which answers an open problem.
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