Hypoellipticity and Vanishing Theorems
by Gerardo Mendoza  

Vol. 8 No. 2 (2013) P.231~P.258

ABSTRACT

Let $-i\mathcal{L}_\mathcal{T}$ (essentially Lie derivative with respect to $\mathcal{T}$, a smooth nowhere zero real vector field) and $P$ be commuting differential operators, respectively of orders $1$ and $m\geq 1$, the latter formally normal, both acting on sections of a vector bundle over a closed manifold. It is shown that if $P+(-i\mathcal{L}_\mathcal{T})^m$ is elliptic then the restriction of $-i\mathcal{L}_\mathcal{T}$ to $\mathscr{D}\subset \ker P\subset L^2$ ($\mathscr{D}$ is carefully specified) yields a selfadjoint operator $-i\mathcal{L}_\mathcal{T}|_\mathscr{D}:\mathscr{D}\subset\ker P\to \ker P$ with compact resolvent. It is also shown that, in the presence of an additional hypothesis on microlocal hypoellipticity of $P$, $-i\mathcal{L}_\mathcal{T}|_\mathscr{D}$ is semi-bounded. These results are applied to CR manifolds on which $\mathcal{T}$ acts as an infinitesimal CR transformation which are then shown to yield versions of Kodaira's vanishing theorem.

KEYWORDS
Spectral theory, hypoellipticity, CR manifolds, cohomology, vanishing theorems

MATHEMATICAL SUBJECT CLASSIFICATION 2010
Primary: 58C4, 32L20, 32V05, 58J10

MILESTONES

Received: 2012-12-29
Revised : 2013-04-20
Accepted: 2013-04-21

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