On the singularities of the Szeg\"o projections on lower energy forms

Let $X$ be an abstract not necessarily compact orientable CR manifold of dimension $2n-1$, $n⩾2$. Let ${\mathrm{\square }}_b^{(q)}$ be the Gaffney extension of Kohn Laplacian for $(0,q)$-forms. We show that the spectral function of ${\mathrm{\square }}_b^{(q)}$ admits a full asymptotic expansion on the non-degenerate part of the Levi form. As a corollary, we deduce that if $X$ is compact and the Levi form is non-degenerate of constant signature on $X$, then the spectrum of ${\mathrm{\square }}_b^{(q)}$ in $]0,\mathrm{\infty }[$ consists of point eigenvalues of finite multiplicity. Moreover, we show that a certain microlocal conjugation of the associated Szeg"o kernel admits an asymptotic expansion under a local closed range condition. As applications, we establish the Szeg"o kernel asymptotic expansions on some weakly pseudoconvex CR manifolds and on CR manifolds with transversal CR $S^1$ actions. By using these asymptotics, we establish some local embedding theorems on CR manifolds and we give an analytic proof of a theorem of Lempert asserting that a compact strictly pseudoconvex CR manifold of dimension three with a transversal CR $S^1$ action can be CR embedded into ${\mathbb{C}}^N$, for some $N\in \mathbb{N}$.