Extension Of Holomorphic Functions Research Articles

Strict quantization of coadjoint orbits

For every semisimple coadjoint orbit \hat{\mathcal{O}} of a complex connected semisimple Lie group \hat{G} , we obtain a family of \hat{G} -invariant products \hat{*}_\hbar on the space of holomorphic functions on \hat{\mathcal{O}} . For every semisimple coadjoint orbit \mathcal{O} of a real connected semisimple Lie group G , we obtain a family of G -invariant products *_\hbar on a space \mathcal{A}(\mathcal{O}) of certain analytic functions on \mathcal{O} by restriction. \mathcal{A}(\mathcal{O}) , endowed with one of the products *_\hbar , is a G -Fréchet algebra, and the formal expansion of the products around \hbar=0 determines a formal deformation quantization of \mathcal{O} , which is of Wick type if G is compact. Our construction relies on an explicit computation of the canonical element of the Shapovalov pairing between generalized Verma modules and complex analytic results on the extension of holomorphic functions.

Characterizations of plurisubharmonic functions

We give characterizations of (quasi-)plurisubharmonic functions in terms of $L^p$-estimates of $\bar\partial$ and $L^p$-extensions of holomorphic functions.

Holomorphic extension of functions along finite families of complex straight lines in an n-circular domain

We consider the continuous functions on the boundary of a bounded n-circular domain D in ℂn, n > 1, which admit one-dimensional holomorphic extension along a family of complex straight lines passing through finitely many points of D. The question is addressed of the existence of a holomorphic extension of these functions to D.

Extension of holomorphic functions defined on singular analytic spaces with growth estimates

Extension of holomorphic functions defined on singular analytic spaces with growth estimates

On the holomorphic extension into a neighborhood of the edge of a wedge in nongeneral position

We generalize some theorems on the possibility of a holomorphic extension of functions into a neighborhood of the edge of a wedge to the case of a two-sided n-circular wedge in nongeneral position (in which a wedge in general position cannot be inscribed).

A Remark on Extensions of CR Functions from Hyperplanes

AbstractIn the characterization of the range of the Radon transform, one encounters the problem of the holomorphic extension of functions defined on ℝ2 \ Δℝ (where Δℝ is the diagonal in ℝ2) and which extend as “separately holomorphic” functions of their two arguments. In particular, these functions extend in fact to ℂ2 \ Δℂ where Δℂ is the complexification of Δℝ. We take this theorem from the integral geometry and put it in the more natural context of the CR geometry where it accepts an easier proof and amore general statement. In this new setting it becomes a variant of the celebrated “edge of the wedge” theorem of Ajrapetyan and Henkin.

Retracts in Polydisk and Analytic Varieties with the H ∞-Extension Property

This article mainly concerns retracts in polydisk, analytic varieties with the H∞-extension property and the three-point Pick problem on \(\mathbb{D}^{3}\) . Arising in the study of Nevanlinna-Pick interpolation on the bidisk, Agler and McCarthy recently discovered a remarkable theorem which characterizes subsets in the bidisk with the polynomial extension property, and in this case, these subsets are retracts. To study H∞-extensions of holomorphic functions from subvarieties of polydisk, one naturally is concerned with retracts in polydisk. Under certain mild assumptions, it is shown that subvarieties with H∞-extension property are exactly retracts. Furthermore, we apply our argument to determine those retracts whose retractions are unique. In particular, a retract in \(\mathbb{D}^{2}\) having at least two different retractions is exactly a balanced disk. As an application, we give a sufficient condition of the uniqueness of the solution for the three-point Pick problem on \(\mathbb{D}^{3}\) .

Extension of Holomorphic Functions From One Side of a Hypersurface

AbstractWe give a new proof of former results by G. Zampieri and the author on extension of holomorphic functions from one side Ω of a real hypersurface M of ℂn in the presence of an analytic disc tangent to M, attached to Ω but not to M. Our method enables us to weaken the regularity assumptions both for the hypersurface and the disc.

On generalized tube domains over

If , are domains over , then Ω = X 1 × X 2 carries a canonical structure of a generalized tube domain over . In response to a question of Jarnicki and Pflug (M. Jarnicki and P. Pflug (2000). Extension of Holomorphic Functions, Springer-Verlag), we show that Ω is one-sheeted and equivalent to a product of convex domains, if it is pseudoconvex.

On Holomorphic Extendability of Functions from Part of the Lie Sphere to the Lie Ball

We describe domains to which the Hua Loo-Keng-type integral for the Lie ball extends holomorphically. We obtain a criterion for the possibility of holomorphic extension of functions from part of the Lie sphere to the Lie ball.

Extension of holomorphic functions through a hypersurface by tangent analytic discs

We prove a new criterion of extendibility of holomorphic functions from a domain Q ⊂ C n by means of analytic discs A tangent to ∂Ω, and verifying ∂A C Ω and ∂A ∩ Ω ≠ > not O.

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.

A counterexample to the Lρ extension of holomorphic functins from subvarieties to pseudoconvex domains *

Using Sibony's example, we get a counterexample to the Lρ (2<p <∞)extension of holomorphic functions from complex linear subspaces to pseudoconvex domains in

Lipschitz and BMO extensions of holomorphic functions from subvarieties to a convex domain

Let be a bounded convex domain with C∞ boundary. Let M be a sub variety in D of dimension one which has no singular points on ∂M and intersects ∂D transversally. Assume that D is weakly totally convex in the complex tangential directions at any point in ∂M We get Lipschitz and BMO extensions of holomorphic functions from M to D

Analytic Discs Attached to Manifolds with Boundary

Let X be a complex manifold of dimension n, M a closed half-space with boundary M, A an analytic disc of X to M, tangent to M at some point z0 of dA n M, and intersecting M + in any neighbourhood of z0. Then holomorphic functions extend from M + to a full neighborhood of z0. This theorem refines the results of [1] where the boundary dA (instead of the whole A) was supposed to intersect M . The argument of the proof consists in constructing a (closed) manifold with boundary W, contained in the envelop of holomorphy of M and such that A c W but A <£ d W. In this situation it is easy to find a new small disc A i c A with 8A l <£ dW. We are therefore in a situation similar to [1], and get the conclusion by exhibiting a disc transversal to d W at z0. Extension of holomorphic functions by the aid of tangent discs attached to M and of 0 is a particular case of a general theorem of wedge extendibility of CR-functions by A. Tumanov; the new part of our theorem is that no assumptions on defect are made. This paper is tightly inspired to the results and the techniques by A. Tumanov [7]. We also owe to A. Tumanov a great help during private communications.

Holomorphic extension of functions from subsets of ?ilov boundaries of circular strictly star-shaped domains

Holomorphic extension of functions from subsets of ?ilov boundaries of circular strictly star-shaped domains

Extension of Holomorphic Functions with Growth Conditions

Extension of Holomorphic Functions with Growth Conditions

Positive-definiteness and its applications to interpolation problems for holomorphic functions

Holomorphic interpolation problems of the Pick-Nevanlinna and Loewner types as well as abstract interpolation theorems on functional Hilbert spaces are considered. Various characterizations are presented for restrictions of bounded holomorphic functions. In addition, certain norm estimates for restrictions and extensions of holomorphic functions are obtained.

On the Extension of Holomorphic Functions with Growth Conditions

On the Extension of Holomorphic Functions with Growth Conditions

On the extension of holomorphic functions on a locally convex space

On the extension of holomorphic functions on a locally convex space