[PDF] Boundary Behavior Of Holomorphic Functions eBook

Author : Elias M. Stein
Publisher : Princeton University Press
Page : 83 pages
File Size : 37,80 MB
Release : 2015-03-08
Category : Mathematics
ISBN : 1400871263

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This book has as its subject the boundary value theory of holomorphic functions in several complex variables, a topic that is just now coming to the forefront of mathematical analysis. For one variable, the topic is classical and rather well understood. In several variables, the necessary understanding of holomorphic functions via partial differential equations has a recent origin, and Professor Stein's book, which emphasizes the potential-theoretic aspects of the boundary value problem, should become the standard work in the field. Originally published in 1972. The Princeton Legacy Library uses the latest print-on-demand technology to again make available previously out-of-print books from the distinguished backlist of Princeton University Press. These editions preserve the original texts of these important books while presenting them in durable paperback and hardcover editions. The goal of the Princeton Legacy Library is to vastly increase access to the rich scholarly heritage found in the thousands of books published by Princeton University Press since its founding in 1905.

Author : Fausto Di Biase
Publisher : Birkhauser
Page : 300 pages
File Size : 21,74 MB
Release : 2006-10
Category : Mathematics
ISBN : 9780817642990

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This monograph examines the boundary behavior of holomorphic functions in several complex variables. Moving beyond the early ideas of Fatou and others, Koranyi and then Stein in the late 1960s and early 1970s deepened the study of Fatou-type theorems in several complex variables, showing that in a general context, approach regions of a shape dramatically larger than non-tangential will give rise to a Fatou-type theorem. These have become known as the admissible regions of Koranyi and Stein. It turns out, however, that the admissible approach regions are only optimal on strongly pseudoconvex domains. Considerable effort has been made in the last 20 years to adapt Fatou theory, and the approach regions in particular, to the Levi geometry of a given domain in multidimensional complex space. The work of Di Biase in the late 1990s is devoted to the Nagel--Stein phenomenon, describing a more general notion of approach region that supersedes the classical ideas of non-tangential and admissible. Krantz's work Function Theory of Several Complex Variables (2000), still the only introduction to the subject, focuses on methods based on maximal function estimates. To date, the main open problem, which is the special focus of this book, is the issue of determining the {it optimal natural approach regions} for the almost everywhere convergence to the boundary of certain smoothly bounded pseudoconvex domains. This book provides the proper framework for the eventual solution of the main problem. This work gives an updated, comprehensive, and self-contained exposition of many results that have never appeared in book form. Starting with foundational material, i.e., from the unit disc in one complexvariable, the reader is lead to the latest discoveries in higher dimensions. New results in boundary value issues of holomorphic functions are examined, which in turn point to new open problems. The book may be used by analysts for individual study or by graduate students.

Author : Baili Min
Publisher :
Page : 70 pages
File Size : 14,86 MB
Release : 2011
Category : Electronic dissertations
ISBN :

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In the theory of several complex variables, the Fatou type problems, the Lindelöf principle, and inner functions have been well studied for strongly pseudoconvex domains. In this thesis, we are going to study more generalized domains, those of finite type. In Chapter 2 we show that there is no Fatou's theorem for approach regions complex tangentially broader than admissible ones, in domains of finite type. In Chapter 3 discussing the Lindelöf principle, we provide some conditions which yield admissible convergence. In Chapter 4 we construct inner functions for a type of domains more general than strongly pseudoconvex ones. Discussion is carried out in C2.

Author : Ryan Keddie Tully-Doyle
Publisher :
Page : 143 pages
File Size : 14,2 MB
Release : 2015
Category :
ISBN : 9781339091945

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This dissertation concerns the investigation of function theoretic properties of certain classes of holomorphic functions in two or more variables by means of operator theoretic methods. Of primary concern will be the Schur class, the class of holomorphic functions from the complex polydisk into the complex unit disk, and the Pick class, the class of holomorphic functions from the complex poly-upperhalfplane into the complex upperhalfplane. In more than two variables, our results will concern certain large subclasses of these functions that satisfy an operator-theoretic condition analogous to a classical inequality of functions of one variable due to von Neumann [vN51]. These subclasses are typically referred to as the Schur-Agler subclass of the Schur functions (introduced in [Agl90]), and the Loewner subclass of the Pick functions (introduced in [AMY12b]. (In one or two variables, these subclasses coincide with the whole class.) These functions are amenable to investigation by means of an operator-theoretic construct called a Hilbert space model, introduced in [Agl90], which relates operator theoretic properties with function theoretic behavior. Hilbert space models are associated with and closely related to the notion of a transfer function realization from engineering and control theory [Hel87]. In Chapter 2, we describe a generalization of Hilbert space models for Schur functions on the bidisk that is well-suited to the investigation of boundary behavior of a function at a class of singular points for the function on the 2-torus. We prove that generalized models with certain regularity properties exist at these singularities. We then solve two function theoretic problems. First, we characterize the directional derivatives of a function in the Schur class at a singular point on the torus where a Caratheodory condition holds (following the generalization of the Julia-Carathedory theorem in [AMY12]. Second, we develop a representation theorem for functions in the two-variable Pick class analogous to the Nevanlinna representation theorem characterizing the Cauchy transforms of positive measures on the real line. In Chapter 3, we investigate more closely the structure of the generalized Hilbert space model. We characterize the directional derivatives in terms of a rational function depending on the structure of a positive contraction associated with a generalized model of a given Schur function. We describe classes of generalized models corresponding to different classes of singular points in the boundary for a Schur function in two variables. In Chapter 4, we generalize to several variables the Nevanlinna representation first investigated in Chapter 2. We show that for the Loewner class, there are representation formulae in terms of densely-defined self-adjoint operators on a Hilbert space that classify completely the Loewner class. We identify four types of such representations, and we obtain function-theoretic conditions that are necessary and sufficient for a given function to possess a representation of each of the four types.