scholarly journals An interpolation, peak and zero set on a weakly pseudoconvex domain

1980 ◽  
Vol 21 ◽  
pp. 175-176
Author(s):  
J. del Castillo
Keyword(s):  
Pseudoconvex Domain ◽  
Zero Set ◽  
Weakly Pseudoconvex ◽  
Weakly Pseudoconvex Domain

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.

L∞-continuity of Henkin operators solving in certain weakly pseudoconvex domains of ℂ2

1984 ◽  
Vol 99 (1-2) ◽  
pp. 25-33 ◽  
Author(s):  
Joan Verdera
Keyword(s):  
Pseudoconvex Domain ◽  
Riesz Potential ◽  
Positive Function ◽  
Type Operator ◽  
Pseudoconvex Domains ◽  
Weakly Pseudoconvex ◽  
Potential Type ◽  
Image Position

SynopsisLet ψ ∈ C2[0,1] be a positive function on (0, 1]. Under certain assumptions on ψ, the setis a pseudoconvex domain with C2-boundary, for which it is possible to construct a Henkin-type operator Hψ = Kψ + Bψ solving in Dψ. The operator Bψ, is L∞-continuous because it has a Riesz potential type kernel, while the L∞-continuity of Kψ depends on the flatness of ψ at 0. Our main result states that Kψ is continuous from L∞(∂Dψ) into L∞(Dψ) if and only if

Functions in the Schwartz algebra that are invertible in the sense of Ehrenpreis

2019 ◽  
Vol 484 (1) ◽  
pp. 7-11
Author(s):  
N. F. Abuzyarova
Keyword(s):  
Entire Function ◽  
Exponential Type ◽  
Zero Set ◽  
Schwartz Algebra

We consider the problem of obtaining the restrictions on the zero set of an entire function of exponential type under which this function belongs to the Schwartz algebra and invertible in the sense of Ehrenpreis.

scholarly journals A note on extensions of multilinear maps defined on multilinear varieties

2021 ◽  
pp. 1-26
Author(s):  
W. T. Gowers ◽  
L. Milićević
Keyword(s):  
Finite Fields ◽  
Vector Spaces ◽  
Restriction Map ◽  
Zero Set ◽  
Dimensional Vector ◽  
Prime Field ◽  
Finite Dimensional ◽  
Multilinear Maps ◽  
Multilinear Map

Abstract Let $G_1, \ldots , G_k$ be finite-dimensional vector spaces over a prime field $\mathbb {F}_p$ . A multilinear variety of codimension at most $d$ is a subset of $G_1 \times \cdots \times G_k$ defined as the zero set of $d$ forms, each of which is multilinear on some subset of the coordinates. A map $\phi$ defined on a multilinear variety $B$ is multilinear if for each coordinate $c$ and all choices of $x_i \in G_i$ , $i\not =c$ , the restriction map $y \mapsto \phi (x_1, \ldots , x_{c-1}, y, x_{c+1}, \ldots , x_k)$ is linear where defined. In this note, we show that a multilinear map defined on a multilinear variety of codimension at most $d$ coincides on a multilinear variety of codimension $O_{k}(d^{O_{k}(1)})$ with a multilinear map defined on the whole of $G_1\times \cdots \times G_k$ . Additionally, in the case of general finite fields, we deduce similar (but slightly weaker) results.

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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