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

TL;DR

This paper studies the asymptotic behavior of Szeg"o projections on CR manifolds, revealing spectral properties and providing kernel expansions that lead to embedding theorems and insights into the geometry of these manifolds.

Contribution

It establishes asymptotic expansions for the spectral function and Szeg"o kernel on CR manifolds, and applies these results to embedding theorems and spectral analysis.

Findings

Spectral function admits a full asymptotic expansion on non-degenerate Levi form regions.

Spectrum of b in positive reals consists of finite-multiplicity eigenvalues for compact manifolds.

Szeg"o kernel admits asymptotic expansion under local closed range condition.

Abstract

Let $X$ be an abstract not necessarily compact orientable CR manifold of dimension $2n-1$, $n\geqslant2$. Let $\Box^{(q)}_{b}$ be the Gaffney extension of Kohn Laplacian for $(0,q)$-forms. We show that the spectral function of $\Box^{(q)}_b$ 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 $\Box^{(q)}_b$ in $]0,\infty[$ 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 $N\in\mathbb N$.