Joseph J. Kohn
Multipliers on Pseudoconvex Domains with Real Analytic Boundaries
Preprint series: ESI preprints

MSC:

32F15 Pseudoconvex domains

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}

Abstract: This paper is concerned with (weakly) pseudoconvex real analytic hypersurfaces
in $\mathbb C^n$. We are motivated by the study of local boundary regularity
of the $\bar\partial$-Neumann problem. Subelliptic estimates in a neighborhood
of a point P in the boundary (which imply regularity) are controlled by ideals
of germs of real analytic functions $I^1(P),\dots,I^{n-1}(P)$. These ideals have
the property that a subelliptic estimate holds for $(p,q)$-forms in a neighborhood
of P if and only if $1\in I^q(P)$. The geometrical meaning of this is that
$1\in I^q(P)$ if and only if there is a neighborhood of P such that there does
not exist a q-dimensional complex analytic manifold contained in the intersection
of this neighborhood. Here we present a method to construct these
manifolds explicitly. That is, if $1\notin I^q(P)$ then in every neighborhood of
P we give an explicit construction of such a manifold. This result is part of a
program to give a more precise understanding of regularity in terms of various
norms. The techniques should also be useful in the study of other systems of
partial differential equations.