Domains of holomorphy with edges and lower dimensional boundary singularities

Necessary and sufficient geometric conditions are given for domains with regular boundary points and edges to be domains of holomorphy provided the remainder boundary subset is of zero Hausdorff 1-codimensional measure.Comment: 12 page

Extension of CR-functions on wedges

Given a wedge V with generic edge in a submanifold M of $C^N$, in general, CR-functions on V do not extend to a wedge in $C^N$ in the direction of each nontrivial value of the Levi form of M as was recently observed by Eastwood and Graham. In this paper we propose an invariant geometric way of selecting those Levi form directions that are responsible for the extension. These are the Levi form directions of the complex tangent vectors, whose complex planes intersect the tangent cone to the wedge in angles larger than $\pi/2$. We show by several examples the optimality of this criterion. Applications to regularity of CR-functions are also given generalizing the edge-of-the-wedge theorem of Ajrapetyan-Henkin

CRCR Extension from manifolds of higher type

In this paper, a generalization of the "sector property" theorem first pioneered by Baouendi, Rothschild and Treves is given. The main contribution consists in showing that if a submanifold of \C^n with higher codimension is locally presented in a weighted normal form, similar to that described by Bloom and Graham, then a characterization is given as to when a vector in the tangent space belongs to the analytic wave front set for the set of locally defined CR functions on the submanifold. This characterization is described in terms of the sign of the inner product of this vector with the local graphing functions for the submanifold on sectors of suitable size along complex lines in its tangent space. Examples are given to show that under certain circumstances, the results are sharp. Previous results by Baouendi et. al. contained a semi-rigidity assumption which is not assumed in the present paper. The hypoanalytic wave front set determines the cone of directions in which CR functions extend analytically to the ambient space and thus provides an explicit description of the local hull of holomorphy of a submanifold of \C^n.Comment: 22 page

Complex Manifolds In QQ-Convex Boundaries

We consider a smooth boundary b\Omega which is q-convex in the sense that its Levi-form has positive trace on every complex q-plane. We prove that b\Omega is tangent of infinite order to the complexification of each of its submanifolds which is complex tangential and of finite bracket type. This generalizes Diederich-Fornaess [Annals 1978] from pseudoconvex to q-convex domains. We also readily prove that the rows of the Levi-form are (1/2)-subelliptic multipliers for the di-bar-Neumann problem on q-forms (cf. Ho [Math. Ann. 1991]). This allows to run the Kohn algorithm of [Acta Math. 1979] in the chain of ideals of subelliptic multipliers for q-forms. If b\Omega is real analytic and the algorithm stucks on q-forms, then it produces a variety of holomorphic dimension q, and in fact, by our result above, a complex q-manifold which is not only tangent but indeed contained in b\Omega. Altogether, the absence of complex q-manifolds in b\Omega produces a subelliptic estimate on q-forms

The Diederich-Fornaess index and the global regularity of the di-bar-Neumann problem

We describe along the guidelines of Kohn "Quantitative estimates..." (1999), the constant E_s which is needed to control the commutator of a totally real vector field T with di-bar* in order to have Sobolev s-regularity of the Bergman projection in any degree of forms, on a smooth pseudoconvex domain D of the complex space. This statement, not explicit in Kohn's paper, yields Straube's Theorem in "A sufficient condition..." (2008). Next, we refine the pseudodifferential calculus at the boundary in order to relate, for a defining function r of D, the operators (T^+)^{-delta/2} and (-r)^{delta/2}. We are thus able to extend to general degree of forms the conclusion of Kohn which only holds for functions: if for the Diederich-Fornaess index delta of D, we have that (1-\delta)^{1/2} < E_s, then the Bergman projection is s-regular

Gain/Loss of derivatives for complex vector fields

In \C_z\times\R_t we consider the function $g=g(z)$, set g_1=\di_z g, g_{1\bar 1}=\di_z\dib_zg and define the operator L_g=\di_z+ig_1\di_t. We discuss estimates with loss of derivatives, in the sense of Kohn, for the system $(\bar L_g,f^kL_g)$ where $(\bar L_g,L_g)$ is $\frac1{2m}$ subelliptic at 0 and $f(0)=0,\,\,df(0)\neq0$. We prove estimates with a loss $l=\frac{k-1}{2m}$ if the "multiplier" condition |f|\simgeq |g_{1\bar 1}|^{\frac1{2(m-1)}} is fulfilled. (For estimates without cut-off, subellipticity can be weakened to compactness and this results in a loss of $l=\frac [{2(m-1)}$.) For the choice $(g,f^k)=(|z|^{2m},\bar z^k)$ this result was obtained by Kohn and Bove-Derridj-Kohn-Tartakoff for $m=1$ and $m\geq1$ respectively. Also, the loss $l=\frac{k-1}{2m}$ was proven to be optimal. We show that it remains optimal for the model $(g,f^k)=(x^{2m},x^k)$. Instead, for the model $(g,f^k)=(|z|^{2m},x^k)$, in which the multiplier condition is violated, the loss is not lowered by the type and must be $\geq \frac{k-1}2$.Comment: The paper contains an error in the proof of Theorem 2.5: estimate (2.16) (a) is incorrect. This misses the proof of the optimality of the loss of derivatives which was a major point of the submissio

Hypoellipticity of the ∂ˉ\bar\partial-Neumann problem at exponentially degenerate points

We prove that the $\bar\partial$-Neumann solution operator is locally regular in a domain which has compactness estimates, is of finite type outside a curve transversal to the CR directions and for which the holomorphic tangential derivatives of a defining function are subelliptic multipliers in the sense of Kohn.Comment: 11 pages. Revision of arxiv 10040919v1, April 6 201

Precise subelliptic estimates for a class of special domains

For the $\bar\partial$-Neumann problem on a regular coordinate domain \Omega\subset \C^{n+1}, we prove $\epsilon$-subelliptic estimates for an index $\epsilon$ which is in some cases better than $\epsilon=\frac1{2m}$ ($m$ being the {\it multiplicity}) as it was previously proved by Catlin and Cho in \cite{CC08}. This also supplies a much simplified proof of the existing literature. Our approach is founded on the method by Catlin in \cite{C87} which consists in constructing a family of weights $\{\phi^\delta\}$ whose Levi form is bigger than $\delta^{-2\epsilon}$ on the $\delta$-strip around $\partial\Omega$.Comment: 9 page

Regularity at the Boundary and Tangential Regularity

For a pseudoconvex domain in complex space, we prove the equivalence of the local hypoellipticity of the system (di-bar, di-bar*) with the system (di-bar_b,di-bar*_b) induced in the boundary. This develops a result of ours which used the theory of the "harmonic" extension by Kohn. This technique is inadequate for the purpose of the present paper and must be replaced by the "holomorphic" extension introduced by the authors in former work

Subellipticity of the ∂ˉ\bar\partial-Neumann problem on a weakly qq-pseudoconvex/concave domain

For a domain $D$ of $\mathbb{C}^n$ which is weakly $q$-pseudoconvex or $q$-pseudoconcave we give a sufficient condition for subelliptic estimates for the $\bar{\partial}$-Neumann problem. The paper extends to domains which are not necessarily pseudoconvex, the results and the techniques of Catlin. MSC: 32D10, 32U05, 32V2