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.

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

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 Extension of holomorphic L2-functions with weighted growth conditions

1992 ◽  
Vol 126 ◽  
pp. 141-157 ◽  
Author(s):  
Klas Diederich ◽  
Gregor Herbort
Keyword(s):  
Fixed Point ◽  
Holomorphic Function ◽  
Lebesgue Measure ◽  
Complex Plane ◽  
Pseudoconvex Domain ◽  
Smooth Boundary ◽  
Small Neighborhood ◽  
Growth Conditions

In this article a new contribution to the following question is given: Let Ω ⊂ ⊂ Cn be a bounded pseudoconvex domain with C∞-smooth boundary, q ∈ ∂Ω a fixed point and H a k-dimensional affine complex plane such that q ∈ H and H intersects ∂Ω at q transversally. Let U be a suitably small neighborhood of q, and denote by r a C∞-defining function of Ω on U. Under which conditions on ∂Ω near q is it possible to find an exponent η>0 > 0 such that every holomorphic function f on Ω′ = H ∩Ω∩ U withwhere dλ′ denotes the Lebesgue-measure on H, can be extended to a holomorphic function ^f on Ω ∩ U such that even

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 $i\varphi \partial \bar \partial_\rho$.

scholarly journals Variétés Singulières et Extension des Fonctions Holomorphes

2008 ◽  
Vol 192 ◽  
pp. 151-167
Author(s):  
Vincent Duquenoy ◽  
Emmanuel Mazzilli
Keyword(s):  
Pseudoconvex Domain ◽  
Holomorphic Functions ◽  
Strictly Pseudoconvex Domain

AbstractIn this paper, we study a problem of extension of holomorphic functions given on a complex hypersurface with singularities on the boundary of a strictly pseudoconvex domain.

scholarly journals A remark on the theorem of Ohsawa-Takegoshi

2000 ◽  
Vol 158 ◽  
pp. 185-189 ◽  
Author(s):  
Klas Diederich ◽  
Emmanuel Mazzilli
Keyword(s):  
Pseudoconvex Domain ◽  
Complex Analysis ◽  
Holomorphic Functions ◽  
Growth Conditions ◽  
Analytic Subset ◽  
Famous Theorem ◽  
Restriction Operator

If D ⊂ ℂn is a pseudoconvex domain and X ⊂ D a closed analytic subset, the famous theorem B of Cartan-Serre asserts, that the restriction operator r : (D) → (X) mapping each function F to its restriction F|X is surjective. A very important question of modern complex analysis is to ask what happens to this result if certain growth conditions for the holomorphic functions on D and on X are added.

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].

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