scholarly journals Extending bounded holomorphic functions from certain subvarieties of a weakly pseudoconvex domain

1984 ◽  
Vol 110 (1) ◽  
pp. 9-19 ◽  
Author(s):  
K. Adachi
Keyword(s):  
Pseudoconvex Domain ◽  
Holomorphic Functions ◽  
Weakly Pseudoconvex ◽  
Weakly Pseudoconvex Domain ◽  
Bounded Holomorphic

scholarly journals SEMIGLOBAL EXTENSION OF MAXIMALLY COMPLEX SUBMANIFOLDS

2011 ◽  
Vol 84 (3) ◽  
pp. 458-474
Author(s):  
GIUSEPPE DELLA SALA ◽  
ALBERTO SARACCO
Keyword(s):  
Pseudoconvex Domain ◽  
Complex Submanifold ◽  
Weakly Pseudoconvex ◽  
Complex Variety ◽  
Complex Submanifolds ◽  
Weakly Pseudoconvex Domain ◽  
Analytic Sets

AbstractLet A be a domain of the boundary of a (weakly) pseudoconvex domain Ω of ℂn and M a smooth, closed, maximally complex submanifold of A. We find a subdomain E of ℂn, depending only on Ω and A, and a complex variety W⊂E such that bW=M in E. Moreover, a generalization to analytic sets of depth at least 4 is given.

scholarly journals Toeplitz operators on certain weakly pseudoconvex domains

1981 ◽  
Vol 31 (1) ◽  
pp. 1-14 ◽  
Author(s):  
David Crocker ◽  
Iain Raeburn
Keyword(s):  
Hardy Space ◽  
Pseudoconvex Domain ◽  
Toeplitz Operators ◽  
Pseudoconvex Domains ◽  
Main Interest ◽  
Weakly Pseudoconvex ◽  
Weakly Pseudoconvex Domain ◽  
Continuous Symbol ◽  
Image Position ◽  
The Ideal

AbstractLet Ω be the weakly pseudoconvex domainand let ∂Ω be its boundary. If ϕ ∈ L∞ (∂Ω), we denote by Tϕ, the Toephtz operator with symbol ϕ acting on the Hardy space H2(∂Ω), and by J(∂Ω) the C*-subalgebra of B(H2(∂Ω)) generated by the Toeplitz operators with continuous symbol. Our main theorem asserts that J(∂Ω) contains the ideal K of all compact operators on H2(∂Ω), and that the symbol map ϕ→Tϕ induces an isomorphism of C(∂Ω) onto the quotient C*-algebra ℑ(∂Ω)/K. Similar results have been established before for other domains, and in particular when Ω is strongly pseudoconvex. The main interest of our results lies in their proofs: ours are elementary, whereas those used in the strongly pseudoconvex case depend heavily on the theory of the tangential Cauchy-Riemann operator.

DENSE STABLE RANK AND RUNGE-TYPE APPROXIMATION THEOREMS FOR MAPS

2021 ◽  
pp. 1-29
Author(s):  
ALEXANDER BRUDNYI
Keyword(s):  
Disjoint Union ◽  
A Priori ◽  
Holomorphic Functions ◽  
Stable Rank ◽  
Open Unit ◽  
Uniform Estimates ◽  
Approximation Theorems ◽  
Type Approximation ◽  
Similar Approximation ◽  
Bounded Holomorphic

Abstract Let $H^\infty ({\mathbb {D}}\times {\mathbb {N}})$ be the Banach algebra of bounded holomorphic functions defined on the disjoint union of countably many copies of the open unit disk ${\mathbb {D}}\subset {{\mathbb C}}$ . We show that the dense stable rank of $H^\infty ({\mathbb {D}}\times {\mathbb {N}})$ is $1$ and, using this fact, prove some nonlinear Runge-type approximation theorems for $H^\infty ({\mathbb {D}}\times {\mathbb {N}})$ maps. Then we apply these results to obtain a priori uniform estimates of norms of approximating maps in similar approximation problems for the algebra $H^\infty ({\mathbb {D}})$ .

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.

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