Klaus Gansberger
On the Weighted $\overline\partial$-Neumann Problem on Unbounded Domains
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}

46E35 Sobolev spaces and other spaces of ``smooth'' functions, embedding theorems, trace theorems

Abstract: Let $\Omega$ be an unbounded, pseudoconvex domain in $\Bbb C^n$
and let $\varphi$ be a $\mathcal C^2$-weight function plurisubharmonic
on $\Omega$. We show both necessary and sufficient conditions for
existence and compactness of a weighted $\dquer$-Neumann operator
$N_\varphi$ on the space $L^2_{(0,1)}(\Omega,e^{-\varphi})$ in terms
of the eigenvalues of the complex Hessian
$(\partial ^2\varphi/\partial z_j\partial\overline z _k)_{j,k}$ of the
weight. We also give some applications to the unweighted $\dquer$-Neumann
problem on unbounded domains.


Keywords: $\overline\partial$-Neumann problem, Sobolev spaces, compactness