Multipliers on Pseudoconvex Domains with Real Analytic Boundaries

Kohn, Joseph J.

Bollettino della Unione matematica italiana, Série 9, Tome 3 (2010) no. 2, pp. 309-324

Résumé

This paper is concerned with (weakly) pseudoconvex real analytic hypersurfaces in $\mathbf{C}^{n}$. 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 $I^{1}(P),\cdots, I^{n-1}(P)$. 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.