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 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.