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.

我们在一个和几个复杂变量中为法图理论奠定了基础。我们描述了EM Stein的《全纯函数的边界行为》一书中的主要贡献。，出版于1972年，至今仍是灵感的来源。我们还说明了他对谐波函数边界行为的研究的贡献。本文的重点不只是简单地阐述众所周知的想法。而是，我们完全重组了主题，以便发挥EM Stein对研究一个或多个变量的全纯和调和函数的边界行为的深刻贡献。在附录中，我们提供了与积分微分有关的新结果的独立证明，正如斯坦因的著作，尤其是上述书中所证明的那样，该主题与谐波的边界行为有着深厚的联系。和全纯函数。