Morse inequalities for Fourier components of Kohn-Rossi cohomology of CR manifolds with $S^1$-action

Let $X$ be a compact connected CR manifold of dimension $2n-1,n\ge 2$ with a transversal CR $S^1$-action on $X$. We study the Fourier components of the Kohn-Rossi cohomology with respect to the $S^1$-action. By studying the Szeg"o kernel of the Fourier components we establish the Morse inequalities on $X$. Using the Morse inequalities we have established on $X$ we prove that there are abundant CR functions on $X$ when $X$ is weakly pseudoconvex and strongly pseudoconvex at a point.