Phillip S. Harrington

Regularity Results for on CR-Manifolds of Hypersurface Type

We introduce a class of CR-manifolds of hypersurface type called weak Y(q)-manifolds that includes Y(q) manifolds and q-pseudoconvex manifolds. We develop the L 2-regularity theory of the complex Green operator on weak Y(q) manifolds and show that and the Kohn Laplacian have closed range at all Sobolev levels, the space of harmonic forms is finite dimensional, the Szegö kernel is continuous and can be solved in C ∞ on the appropriate forms levels. Our argument involves building a weighted norm from a microlocal decomposition.

DEFINING FUNCTIONS FOR UNBOUNDED C m DOMAINS

For a domain

mapping properties for the Cauchy–Riemann equations on Lipschitz domains admitting subelliptic estimates

\Omega\subset\mathbb R^n

Bounded Plurisubharmonic Exhaustion Functions for Lipschitz Pseudoconvex Domains in \({\mathbb {C}}{\mathbb {P}}^n\)

, we introduce the concept of a uniformly

A remark on boundary estimates on unbounded Z(q) domains in

C^m

WHAT IS...a CR Submanifold

defining function. We characterize uniformly

Regularity equivalence of the Szegö projection and the complex Green operator

C^m

Global regularity for the ∂¯-Neumann operator and bounded plurisubharmonic exhaustion functions

defining functions in terms of the signed distance function for the boundary and provide a large class of examples of unbounded domains with uniformly

Closed Range for

C^m

\bar\partial

defining functions. Some of our results extend results from the bounded case.