Foundations of Fatou Theory and a Tribute to the Work of E. M. Stein on Boundary Behavior of Holomorphic Functions

We lay the foundations of Fatou theory in one and several complex variables. We describe the main contributions contained in E. M. Stein’s book Boundary Behavior of Holomorphic Functions , published in 1972 and still a source of inspiration. We also give an account of his contributions to the study...

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Bibliographic Details
Published in:The Journal of geometric analysis Vol. 31; no. 7; pp. 7184 - 7296
Main Authors: Di Biase, Fausto, Krantz, Steven G.
Format: Journal Article
Language:English
Published: New York Springer US 01.07.2021
Springer Nature B.V
Subjects:
Abstract Harmonic Analysis
Analytic functions
Complex variables
Convex and Discrete Geometry
Differential Geometry
Dynamical Systems and Ergodic Theory
Elias M. Stein: In Memoriam
Foundations
Fourier Analysis
Geometry
Global Analysis and Analysis on Manifolds
Harmonic functions
Mathematical analysis
Mathematics
Mathematics and Statistics
31B25
32A35
28A15
Almost everywhere convergence
Theorems of Fatou type
32A40
Approach regions
Maximal functions
ISSN:1050-6926, 1559-002X
Online Access:Get full text
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Summary:We lay the foundations of Fatou theory in one and several complex variables. We describe the main contributions contained in E. M. Stein’s book Boundary Behavior of Holomorphic Functions , published in 1972 and still a source of inspiration. We also give an account of his contributions to the study of the boundary behavior of harmonic functions. The point of this paper is not simply to exposit well-known ideas. Rather, we completely reorganize the subject in order to bring out the profound contributions of E. M. Stein to the study of the boundary behavior both of holomorphic and harmonic functions in one and several variables. In an appendix, we provide a self-contained proof of a new result which is relevant to the differentiation of integrals, a topic which, as witnessed in Stein’s work, and especially by the aforementioned book, has deep connections with the boundary behavior of harmonic and holomorphic functions.
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ISSN:1050-6926
1559-002X
DOI:10.1007/s12220-021-00618-z