Asymptotic Expansions of Invariant Metrics of Strictly Pseudoconvex Domains

1995 ◽  
Vol 38 (2) ◽  
pp. 196-206 ◽  
Author(s):  
Siqi Fu
Keyword(s):  
Continuous Function ◽  
Asymptotic Expansions ◽  
Pseudoconvex Domain ◽  
Smooth Boundary ◽  
Pseudoconvex Domains ◽  
Invariant Metrics ◽  
Kobayashi Metric ◽  
Signed Distance ◽  
Strictly Pseudoconvex Domain ◽  
Kobayashi Metrics

AbstractIn this paper we obtain the asymptotic expansions of the Carathéodory and Kobayashi metrics of strictly pseudoconvex domains with C∞ smooth boundaries in ℂn. The main result of this paper can be stated as following:Main Theorem. Let Ω be a strictly pseudoconvex domain with C∞ smooth boundary. Let FΩ(z,X) be either the Carathéodory or the Kobayashi metric of Ω. Let δ(z) be the signed distance from z to ∂Ω with δ(z) < 0 for z ∊ Ω and δ(z) ≥ 0 for z ∉ Ω. Then there exist a neighborhood U of ∂Ω, a constant C > 0, and a continuous function C(z,X):(U ∩ Ω) × ℂn -> ℝ such that and|C(z,X)| ≤ C|X| for z ∊ U ∩ Ω and X ∊ ℂn

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

2003 ◽  
Vol 172 ◽  
pp. 103-110
Author(s):  
Kenzō Adachi
Keyword(s):  
Holomorphic Function ◽  
Pseudoconvex Domain ◽  
Smooth Boundary ◽  
Holomorphic Functions ◽  
Pseudoconvex Domains ◽  
Strictly Pseudoconvex Domain

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.

scholarly journals Estimates of Invariant Metrics on Pseudoconvex Domains of Finite Type inC3

2014 ◽  
Vol 2014 ◽  
pp. 1-7
Author(s):  
Sanghyun Cho ◽  
Young Hwan You
Keyword(s):  
Finite Type ◽  
Pseudoconvex Domain ◽  
Pseudoconvex Domains ◽  
Invariant Metrics ◽  
Small Constant ◽  
Kobayashi Metrics

LetΩbe a smoothly bounded pseudoconvex domain inC3and assume thatz0∈bΩis a point of finite 1-type in the sense of D’Angelo. Then, there are an admissible curveΓ⊂Ω∪{z0}, connecting points  q0∈Ωandz0∈bΩ, and a quantityM(z,X), alongz∈Γ, which bounds from above and below the Bergman, Caratheodory, and Kobayashi metrics in a small constant and large constant sense.

Hankel Operators on Pseudoconvex Domains of Finite Type in ℂ2

1998 ◽  
Vol 50 (3) ◽  
pp. 658-672 ◽  
Author(s):  
Frédéric Symesak
Keyword(s):  
Hardy Space ◽  
Finite Type ◽  
Pseudoconvex Domain ◽  
Bergman Spaces ◽  
Hankel Operators ◽  
Weighted Bergman Spaces ◽  
Pseudoconvex Domains ◽  
Sufficient Condition ◽  
Schatten Class ◽  
Strictly Pseudoconvex Domain

AbstractThe aimof this paper is to study small Hankel operators h on the Hardy space or on weighted Bergman spaces,where Ω is a finite type domain in ℂ2 or a strictly pseudoconvex domain in ℂn. We give a sufficient condition on the symbol ƒ so that h belongs to the Schatten class Sp, 1 ≤ p < +∞.

scholarly journals Toeplitz Algebras on Strongly Pseudoconvex Domains

2007 ◽  
Vol 185 ◽  
pp. 171-186 ◽  
Author(s):  
Guangfu Cao
Keyword(s):  
Continuous Function ◽  
Pseudoconvex Domain ◽  
Function Algebra ◽  
Pseudoconvex Domains ◽  
Chern Classes ◽  
Toeplitz Algebra ◽  
Cohomology Groups ◽  
Ring Isomorphism ◽  
Strongly Pseudoconvex Domains ◽  
Strongly Pseudoconvex Domain

AbstractIn the present paper, it is proved that theK0-group of a Toeplitz algebra on any strongly pseudoconvex domain is always isomorphic to theK0-group of the relative continuous function algebra, and is thus isomorphic to the topologicalK0-group of the boundary of the relative domain. Further there exists a ring isomorphism between theK0-groups of Toeplitz algebras and the Chern classes of the relative boundaries of strongly pseudoconvex domains. As applications of our main result,K-groups of Toeplitz algebras on some special strongly pseudoconvex domains are computed. Our results show that the Toeplitz algebras on strongly pseudoconvex domains have rich structures, which deeply depend on the topological structures of relative domains. In addition, the first cohomology groups of Toeplitz algebras are also computed.

The $\bar \partial $-problem on q-pseudoconvex domains with applications

2013 ◽  
Vol 63 (3) ◽  
Author(s):  
S. Saber
Keyword(s):  
Pseudoconvex Domain ◽  
Smooth Boundary ◽  
Pseudoconvex Domains ◽  
Lipschitz Boundary ◽  
Cauchy Riemann

AbstractFor a q-pseudoconvex domain Ω in ℂn, 1 ≤ q ≤ n, with Lipschitz boundary, we solve the $\bar \partial $-problem with exact support in Ω. Moreover, we solve the $\bar \partial $-problem with solutions smooth up to the boundary over Ω provided that it has smooth boundary. Applications are given to the solvability of the tangential Cauchy-Riemann equations on the boundary.

scholarly journals Atomic decompositions in weighted Bergman spaces of analytic functions on strictly pseudoconvex domains

Filomat ◽  
2021 ◽  
Vol 35 (8) ◽  
pp. 2545-2563
Author(s):  
Milos Arsenovic
Keyword(s):  
Analytic Functions ◽  
Atomic Decomposition ◽  
Smooth Boundary ◽  
Bergman Spaces ◽  
Real Variable ◽  
Weighted Bergman Spaces ◽  
Pseudoconvex Domains ◽  
Atomic Decompositions ◽  
Strictly Pseudoconvex Domain ◽  
Spaces Of Analytic Functions

We construct an atomic decomposition of the weighted Bergman spaces Ap?(D) (0 < p ? 1, ? > -1) of analytic functions on a bounded strictly pseudoconvex domain D in Cn with smooth boundary. The atoms used are atoms in the real-variable sense.

scholarly journals Approximate formulas for canonical homotopy operators for the $\bar \partial$ complex in strictly pseudoconvex domains

2000 ◽  
Vol 87 (2) ◽  
pp. 251 ◽  
Author(s):  
Mats Andersson ◽  
Jörgen Boo
Keyword(s):  
Pseudoconvex Domain ◽  
Smooth Function ◽  
Pseudoconvex Domains ◽  
Boundary Values ◽  
Orthogonal Projections ◽  
Homotopy Operator ◽  
Strictly Pseudoconvex Domain ◽  
Regularity Results ◽  
Homotopy Operators ◽  
Approximate Formulas

Let $D=\{ \rho <0 \}$ be a smoothly bounded strictly pseudoconvex domain in $\boldsymbol C^n$ and $\rho$ a strictly plurisubharmonic smooth defining function. We construct explicit homotopy operators for the $\bar \partial$ complex, which are approximately equal to the homotopy operators that are canonical with respect to the metric $\Omega = i\varphi(-\rho)\partial \bar \partial \log(1/-\rho)$ and weights $(-\rho)^\alpha$, where $\varphi$ is a strictly positive smooth function. We also obtain an explicit operator which is approximately equal to the canonical homotopy operator for $\bar \partial_b$ on $\partial D$. From the explicit operators we obtain regularity results for these canonical operators, including $C^\infty$ regularity and $L^p$-boundedness for the orthogonal projections onto Ker $\bar \partial$ and Ker $\bar \partial_b$. Previously it has been proved, in the ball case and $\varphi \equiv 1$, that the boundary values of the canonical operators coincide with the values of well-known explicit operators due to Henkin and Skoda et al. Previously Lieb and Range have constructed an explicit homotopy operator which is approximately equal to the canonical operator with respect to the metric $i\varphi \partial \bar \partial_\rho$.

Existence and Hölder continuity to solutions of the complex Monge–Ampère-type equations with measures vanishing on pluripolar subsets

2021 ◽  
Author(s):  
Le Mau Hai ◽  
Vu Van Quan
Keyword(s):  
Pseudoconvex Domain ◽  
Hölder Continuity ◽  
Type Equation ◽  
Holder Continuity ◽  
Continuous Solutions ◽  
Hölder Continuous ◽  
Strictly Pseudoconvex Domain ◽  
Monge Ampere

In this paper, we establish existence of Hölder continuous solutions to the complex Monge–Ampère-type equation with measures vanishing on pluripolar subsets of a bounded strictly pseudoconvex domain [Formula: see text] in [Formula: see text].

Sign in / Sign up

Export Citation Format

Share Document