On the singularities of the Szeg\H{o} projections on lower energy forms

Chin-Yu Hsiao Institute of Mathematics, Academia Sinica, Taiwan George Marinescu University of Cologne, Germany

Complex Variables and Complex Analysis mathscidoc:1803.08003

J. Differential Geom, 107, (1), 83-155, 2017
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\H{o} kernel admits an asymptotic expansion under a local closed range condition. As applications, we establish the Szeg\H{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 $N\in\mathbb N$.
Microlocal Analysis, CR geometry, Szego kernel
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  title={On the singularities of the Szeg\H{o} projections on lower energy forms},
  author={Chin-Yu Hsiao, and George Marinescu},
  booktitle={ J. Differential Geom},
Chin-Yu Hsiao, and George Marinescu. On the singularities of the Szeg\H{o} projections on lower energy forms. 2017. Vol. 107. In J. Differential Geom. pp.83-155. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20180329233019255088996.
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