Lebesgue's Theory of Singular Integrals: Foundations of Modern Harmonic Analysis

This book presents a scholarly English translation of Henri Lebesgue's seminal 1909 paper, Sur les Intégrales Singulières, first published in the Annales de l'Université de Toulouse. This foundational work, long known to specialists in French, is made accessible here to a broader mathematical audience through a clear, faithful and pedagogically enriched translation. In addition to the core translation, the book features an original introductory chapter and an appendix by the translator that trace the historical development of harmonic analysis from classical trigonometric series to the Zygmund-Calderón theory, with mention of Carleson's breakthrough in almost everywhere convergence.

The historical survey outlines the evolution of ideas contributed by Dirichlet, Fejér, Dini, Vallée Poussin, Jordan, Kolmogorov, Young and others, illuminating the pathway from early convergence results to modern singular integral theory. The material is presented in structured chapters and enhanced with diagrams, worked examples and graded exercises, offering an accessible yet rigorous learning experience. Supplementary readings are included to encourage deeper exploration and contextual understanding.

This volume is ideal for advanced undergraduate and graduate students, as well as researchers interested in Fourier analysis, trigonometric series and the early development of harmonic analysis. It provides a rich conceptual bridge between classical analysis and modern harmonic tools. It is particularly well-suited for courses on harmonic analysis and the historical foundations of integration theory. Mathematicians, educators and historians of mathematics will find in this book both a pedagogical resource and a valuable preservation of Lebesgue's contribution to the analytic tradition.