Abstract
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.
Highlights
The difference between the boundary behavior of holomorphic functions and that of harmonic functions becomes much more significant if n > 1, due to a subtler interplay between potential theory and complex analysis
When Stein started his work in this second area, classical potential theory had already reached a high degree of development: It had been axiomatized, or was being axiomatized, by the French school
Some precise and geometrically crafted results on boundary behavior did exist, but most of them, with perhaps the only exceptions given by the work of Privalov and Kouznetzoff (1939) for Lyapunov domains in Rn, Tsuji (1939) for the unit ball in Rn, Tsuji (1944) for Lyapunov domains in Rn, and Calderón (1950) for the upper half-space, were confined to planar domains and to holomorphic function and heavily depended on conformal mappings, which transfer the problem to the unit disc [24,25,144,176,177]
Summary
The difference between the boundary behavior of holomorphic functions (defined on a certain class of bounded domain in Cn) and that of harmonic functions (defined on the same domains, seen as subdomains of R2n) becomes much more significant if n > 1, due to a subtler interplay between potential theory and complex analysis. Some precise and geometrically crafted results on boundary behavior did exist, but most of them, with perhaps the only exceptions given by the work of Privalov and Kouznetzoff (1939) for Lyapunov domains in Rn, Tsuji (1939) for the unit ball in Rn, Tsuji (1944) for Lyapunov domains in Rn, and Calderón (1950) for the upper half-space, were confined to planar domains and to holomorphic function and heavily depended on conformal mappings, which transfer the problem to the unit disc [24,25,144,176,177]. Before we plunge into the field, we introduce some notation that will make the treatment run smoothly, and say a few words on the notion of “boundary property” occurring in these matters
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