Joe J Perez
Subelliptic Boundary Value Problems and the $G$-Fredholm Property
Preprint series: ESI preprints

MSC:

32F20 $\overline\partial$- and $\overline\partial_b$-Neumann problems, See also {35N15}

35N15 $\overline\partial$-Neumann problem and generalizations; formal complexes, See also {32F20 and 58G05}

35H05 Hypoelliptic equations and systems, See also {58Gxx}

43A30 Fourier and Fourier-Stieltjes transforms on nonabelian groups and on semigroups, etc.

Abstract: Let $G$ be a unimodular Lie group, $X$ a compact manifold with boundary,
and $M$ be the total space of a principal bundle $G\to M\to X$ so that
$M$ is also a complex manifold satisfying a local subelliptic estimate.
In this work, we show that if $G$ acts by holomorphic transformations
in $M$, then the Laplacian $\square=\bar\partial^{*}\bar\partial+\bar\partial\bar\partial^{*}$
on $M$ has the following properties: The kernel of $\square$ restricted
to the forms $\Lambda^{p,q}$ with $q>0$ is a closed, $G$-invariant subspace
in $L^{2}(M,\Lambda^{p,q})$ of finite $G$-dimension. Secondly, we show that
if $q>0$, then the image of $\square$ contains a closed, $G$-invariant subspace
of finite codimension in $L^{2}(M,\Lambda^{p,q})$. These two properties taken
together amount to saying that $\square$ is a $G$-Fredholm operator. In similar
circumstances, the boundary Laplacian $\square_b$ has similar properties.


Keywords: $\bar\partial$-Neumann problem, subelliptic operators