Szeg\H{o} kernel asymptotic expansion on CR manifolds with $S^1$ action

Let $X$ be a compact connected strongly pseudoconvex CR manifold of dimension $2n+1,n\ge 1$ with a transversal CR $S^1$ action on $X$. We establish an asymptotic expansion for the $m$-th Fourier component of the Szeg\H{o} kernel function as $m\to \mathrm{\infty }$, where the expansion involves a contribution in terms of a distance function from lower dimensional strata of the $S^1$ action. We also obtain explicit formulas for the first three coefficients of the expansion.