Pseudoconvex Domain In Cn Research Articles

A solution operator for the \overline∂ equation in Sobolev spaces of negative index

Let Ω \Omega be a strictly pseudoconvex domain in C n \mathbb {C}^n with C k + 2 C^{k+2} boundary, k ≥ 1 k \geq 1 . We construct a ∂ ¯ \overline \partial solution operator (depending on k k ) that gains 1 2 \frac 12 derivative in the Sobolev space H s , p ( Ω ) H^{s,p} (\Omega ) for any 1 > p > ∞ 1>p>\infty and s > 1 p − k s>\frac {1}{p} -k . If the domain is C ∞ C^{\infty } , then there exists a ∂ ¯ \overline \partial solution operator that gains 1 2 \frac 12 derivative in H s , p ( Ω ) H^{s,p}(\Omega ) for all s ∈ R s \in \mathbb {R} . We obtain our solution operators via the method of homotopy formula. A novel technique is the construction of “anti-derivative operators” for distributions defined on bounded Lipschitz domains.

On smoothing properties of the Bergman projection

We study smoothing properties of the Bergman projection and weighted Bergman projections. In particular, we relate these properties to the hyperconvexity index of a pseudoconvex domain in Cn. The notion of a hyperconvexity index was first introduced by B.Y. Chen, which provides a flexible criterion for studying geometric properties of hyperconvex domains. We also obtain a new estimate of weighted Bergman projections, which extends a well-known estimate of Berndtsson and Charpentier. We then give some applications of this estimate.

Lorentz boundedness and compactness characterization of integral commutators on spaces of homogeneous type

We study the Lorentz boundedness properties of Calderón-type commutators and relate those properties to the spaces of bounded mean oscillation and vanishing mean oscillation. In application, we obtain Lorentz boundedness and compactness characterization of integral commutators with singular kernel of Calderón–Zygmund type on strictly pseudoconvex domains in ℂn, or convex domain of finite type in ℂn.

Morrey boundedness and compactness characterizations of integral commutators with singular kernel on strictly pseudoconvex domains in [formula omitted

We prove the Morrey boundedness and compactness characterization of integral commutators with singular kernel of Calderón-Zygmund type on pseudo convex domains in Cn. Our results extend some of the ones of Krantz-Li in [16,17].

Some properties of h-extendible domains in [formula omitted

The purpose of this article is twofold. The first aim is to characterize h-extendibility of smoothly bounded pseudoconvex domains in Cn+1 by their noncompact automorphism groups. Our second goal is to show that if the squeezing function tends to 1 at an h-extendible boundary point of a smooth pseudoconvex domain in Cn+1, then this point must be strongly pseudoconvex.

Holomorphic Approximation on Certain Weakly Pseudoconvex Domains in Cn

The purpose of this paper is to study the Mergelyan approximation property in L p and C k -scales on certain weakly pseudoconvex domains of finite/infinite type in C n . At the heart of our results lies the solvability of the additive Cousin problem with bounds as well as estimates of the ∂ ¯ -equation in the corresponding topologies.

Bounded strictly pseudoconvex domains in [formula omitted] with obstruction flat boundary II

On a bounded strictly pseudoconvex domain in Cn, n>1, the smoothness of the Cheng-Yau solution to Fefferman's complex Monge-Ampere equation up to the boundary is obstructed by a local CR invariant of the boundary. For a bounded strictly pseudoconvex domain Ω⊂C2 diffeomorphic to the ball, we prove that the global vanishing of this obstruction implies biholomorphic equivalence to the unit ball, subject to the existence of a holomorphic vector field satisfying a mild approximate tangency condition along the boundary. In particular, by considering the Euler vector field multiplied by i the result applies to all domains in a large C1 open neighborhood of the unit ball in C2. The proof rests on establishing an integral identity involving the CR curvature of ∂Ω for any holomorphic vector field defined in a neighborhood of the boundary. The notion of ambient holomorphic vector field along the CR boundary generalizes naturally to the abstract setting, and the corresponding integral identity still holds in the case of abstract CR 3-manifolds.

On h-extendible domains and associated models

A boundary point of a smooth pseudoconvexdomain in Cn is said to be h-extendible if its Catlin's multi-type coincides with its D'Angelo's multi-type. There is a local model defined by Catlin's multi-weight. In this paper, we show that a domain in Cn with a noncompact automorphism group is biholomorphically equivalent to its associated model if there exists a sequence of automorphisms of the domain that has an orbit converging to an h-extendible boundary point non-tangentially in a cone region.

Iterates of holomorphic self-maps on pseudoconvex domains of finite and infinite type in $\mathbb C^n$

Using the lower bounds on the Kobayashi metric established by the first author, we prove a Wolff-Denjoy-type theorem for a very large class of pseudoconvex domains in Cn. This class includes many pseudoconvex domains of finite type and infinite type.

Maximal hypoellipticity for the [formula omitted]-Neumann problem

We establish maximal hypoellipticity for the ∂¯-Neumann problem on smooth, bounded pseudoconvex domains in Cn under the weakest possible condition on the Levi form. In particular, maximal hypoellipticity holds on the level of (n−1)-forms for pseudoconvex domains of finite commutator type.

Boundary regularity of the solution to the complex Monge-Ampère equation on pseudoconvex domains of infinite type

All rights reserved. Let Ω be a C2-smooth, bounded, pseudoconvex domain in Cn satisfying the -property. The f -property is a consequence of the geometric of the boundary. All pseudoconvex domains of finite type satisfy the f-property as well as many classes of domains of infinite type. In this paper, we prove the existence, uniqueness, and weak Holder-regularity up to the boundary of the solution to the Dirichlet problem for the complex Monge-Ampere equation (Formula Presented) The idea of our proof goes back to Bedford and Taylor [1]. However, the basic geometrical ingredient is based on a recent result by Khanh [12].

Comparison of invariant functions on strongly pseudoconvex domains

It is shown that the Carathéodory distance and the Lempert function are almost the same on any strongly pseudoconvex domain in Cn; in addition, if the boundary is C2+ε-smooth, then n+1 times one of them almost coincides with the Bergman distance.

Coherence and other properties of sheaves in the Kohn algorithm

In the smooth case, we prove quasi-flasqueness for the sheaves of all subelliptic multipliers as well as at each of the steps of the Kohn algorithm on a pseudoconvex domain in ℂn. We use techniques by Jean-Claude Tougeron to show that if the domain has a real-analytic defining function, the modified Kohn algorithm involving generating ideals and taking real radicals only in the ring of real-analytic germs yields quasi-coherent sheaves. This sharpens a result obtained by J. J. Kohn in 1979.

Compactness of the [formula omitted]-Neumann operator and commutators of the Bergman projection with continuous functions

Let Ω be a bounded pseudoconvex domain in Cn,n≥2,0≤p≤n, and 1≤q≤n−1. We show that compactness of the ∂¯-Neumann operator, Np,q+1, on square integrable (p,q+1)-forms is equivalent to compactness of the commutators [Pp,q,z¯j] on square integrable ∂¯-closed (p,q)-forms for 1≤j≤n where Pp,q is the Bergman projection on (p,q)-forms. We also show that compactness of the commutator of the Bergman projection with bounded functions percolates up in the ∂¯-complex on ∂¯-closed forms and square integrable holomorphic forms.

On compactness of the \overline{∂}-Neumann problem and Hankel operators

Let Ω = Ω 1 ∖ Ω ¯ 2 \Omega =\Omega _1\setminus \overline {\Omega }_2 , where Ω 1 \Omega _1 and Ω 2 \Omega _2 are two smooth bounded pseudoconvex domains in C n , n ≥ 3 , \mathbb {C}^n, n\geq 3, such that Ω ¯ 2 ⊂ Ω 1 . \overline {\Omega }_2\subset \Omega _1. Assume that the ∂ ¯ \overline {\partial } -Neumann operator of Ω 1 \Omega _1 is compact and the interior of the Levi-flat points in the boundary of Ω 2 \Omega _2 is not empty (in the relative topology). Then we show that the Hankel operator on Ω \Omega with symbol ϕ , H ϕ Ω , \phi , H^{\Omega }_{\phi }, is compact for every ϕ ∈ C ( Ω ¯ ) \phi \in C(\overline {\Omega }) but the ∂ ¯ \overline {\partial } -Neumann operator on Ω \Omega is not compact.

Domains with Hyperbolic Orbit Accumulation Boundary Points

Let Ω be a smoothly bounded pseudoconvex domain in ℂn satisfying the condition R. Suppose that its Bergman kernel extends to \(\overline{\Omega}\times\overline{\Omega}\) minus the boundary diagonal set as a locally bounded function. In this paper we show that for each hyperbolic orbit accumulation boundary point p, there exists a contraction f∈Aut(Ω) at p. As an application, we show that Ω admits a hyperbolic orbit accumulation boundary point if and only if it is biholomorphically equivalent to a domain defined by a weighted homogeneous polynomial and that Ω is of finite D’Angelo type.

The corona problem with two pieces of data

We study the corona problem on the unit ball in C n \mathbb {C}^n , and more generally on strongly pseudoconvex domains in C n \mathbb {C}^n . When the corona problem has just two pieces of data, and an extra geometric hypothesis is satisfied, then we are able to solve it.

Hankel operators and weak factorization for Hardy–Orlicz spaces

We study the holomorphic Hardy-Orlicz spaces H^Φ(Ω), where Ω is the unit ball or, more generally, a convex domain of finite type or a strictly pseudoconvex domain in Cn . The function Φ is in particular such that H ^1(Ω) ⊂ H^Φ (Ω) ⊂ H ^p (Ω) for some p > 0. We develop for them maximal characterizations, atomic and molecular decompositions. We then prove weak factorization theorems involving the space BMOA(Ω). As a consequence, we characterize those Hankel operators which are bounded from H ^Φ(Ω) into H^1 (Ω).

On the Regularity of the Pluricomplex Green Functions

We study an exterior Dirichlet problem for the homogeneous complex Monge-Ampere equation that arises in the pluripotential theory. As a consequence we prove the C 1, 1 regularity of the plruicomplex Green function for the union of a finite collection of strongly pseudoconvex domains in ℂ n .

Lp extension of holomorphic functions from submanifolds to strictly pseudoconvex domains with non-smooth boundary

AbstractLet D be a bounded strictly pseudoconvex domain in ℂn (with not necessarily smooth boundary) and let X be a submanifold in a neighborhood of . Then any Lp (1 ≥ p < ∞) holomorphic function in X ∩ D can be extended to an Lp holomorphic function in D.