A survey on the extension of holomorphic functions on analytic subvarieties

On the extension of holomorphic functions with growth conditions across analytic subvarieties.

We 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.

Various generalizations of Bochner's theorem on the extension of holomorphic functions over tube domains are considered.It is shown that CR functions on tubes over connected, locally closed, locally starlike subsets of R" uniquely extend to CR functions on almost all of the convex hull of the tube set.A CR extension theorem on maximally stratified real submanifolds of C is proven.The above two theorems are used to show that the CR functions (resp.CR distributions) on tubes over a fairly general class of submanifolds of R" uniquely extend to CR functions (CR distributions) on almost all of the convex hull.0. Introduction.One of the major differences between holomorphic functions of several complex variables as opposed to one complex variable is the property of holomorphic extendability.On every connected open set in C, there exists a holomorphic function / which cannot be extended to a holomorphic function on a larger open set containing .This is no longer true in C" ( > 1).Given a connected open set in C (n > 1), is there a largest open set ', containing , such that every holomorphic function on extends holomorphically to '?In general ' does not exist.However, there exists a "largest" complex manifold S, containing , with the property that every holomorphic function on extends to a unique holomorphic function onS.If we restrict ourselves to special 's, there are many results on the extendability of holomorphic functions.A few examples follow: Theorem (Hartogs).Let be a connected open set in C (n > 1) and let K be a compact set in C such that -K is connected.Then every holomorphic function on -K extends to .Bochner's tube theorem.Let U be a connected open set in R" and t( U) (= U X /R"), the tube over U, be the set of points in C whose real parts belong to U. Then every holomorphic function on t(U) extends to the convex hull of t(U).In the 1940s, Bochner and Martinelli among others showed that if is a connected open set in C (n > 1) with C2 boundary then functions satisfying

The goal of this survey is to describe some recent results concerning the \(L^{2}\) extension of holomorphic sections or cohomology classes with values in vector bundles satisfying weak semi-positivity properties. The results presented here are generalized versions of the Ohsawa–Takegoshi extension theorem, and borrow many techniques from the long series of papers by T. Ohsawa. The recent achievement that we want to point out is that the surjectivity property holds true for restriction morphisms to non necessarily reduced subvarieties, provided these are defined as zero varieties of multiplier ideal sheaves. The new idea involved to approach the existence problem is to make use of \(L^{2}\) approximation in the Bochner-Kodaira technique. The extension results hold under curvature conditions that look pretty optimal. However, a major unsolved problem is to obtain natural (and hopefully best possible) \(L^{2}\) estimates for the extension in the case of non reduced subvarieties—the case when Y has singularities or several irreducible components is also a substantial issue.

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.

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.

LetDbe a bounded strictly pseudoconvex domain in ℂn(with not necessarily smooth boundary) and letXbe a submanifold in a neighborhood of. Then anyLp(1 ≥p&lt; ∞) holomorphic function inX∩Dcan be extended to anLpholomorphic function inD.

Holomorphic interpolation problems of the Pick-Nevanlinna and Loewner types as well as abstract interpolation theorems on functional Hubert 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.

This book presents many of the main developments of the past two decades in the study of real submanifolds in complex space, providing crucial background material for researchers and advanced graduate students. The techniques in this area borrow from real and complex analysis and partial differential equations, as well as from differential, algebraic, and analytical geometry. In turn, these latter areas have been enriched over the years by the study of problems in several complex variables addressed here. The authors, M. Salah Baouendi, Peter Ebenfelt, and Linda Preiss Rothschild, include extensive preliminary material to make the book accessible to nonspecialists. One of the most important topics that the authors address here is the holomorphic extension of functions and mappings that satisfy the tangential Cauchy-Riemann equations on real submanifolds. They present the main results in this area with a novel and self-contained approach. The book also devotes considerable attention to the study of holomorphic mappings between real submanifolds, and proves finite determination of such mappings by their jets under some optimal assumptions. The authors also give a thorough comparison of the various nondegeneracy conditions for manifolds and mappings and present new geometric interpretations of these conditions. Throughout the book, Cauchy-Riemann vector fields and their orbits play a central role and are presented in a setting that is both general and elementary.

𝐿^{𝑝} and 𝐻^{𝑝} extensions of holomorphic functions from subvarieties

On the Extension of Holomorphic Functions with Growth Conditions

Various generalizations of Bochner’s theorem on the extension of holomorphic functions over tube domains are considered. It is shown that CR functions on tubes over connected, locally closed, locally starlike subsets of R n {\textbf {R}^n} uniquely extend to CR functions on almost all of the convex hull of the tube set. A CR extension theorem on maximally stratified real submanifolds of C n {\textbf {C}^n} is proven. The above two theorems are used to show that the CR functions (resp. CR distributions) on tubes over a fairly general class of submanifolds of R n {\textbf {R}^n} uniquely extend to CR functions (CR distributions) on almost all of the convex hull.

On the extension of holomorphic functions on a locally convex space

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 \(\mathbb{D}^{3}\) .

Numerical analysis on 8-problem and extension of holomorphic functions from lower dimensional subvarieties