Wilfredo Urbina-Romero – författare

Authored by a ranking authority in Gaussian harmonic analysis, this book embodies a state-of-the-art entrée at the intersection of two important fields of research: harmonic analysis and  probability. The book is intended for a very diverse audience, from graduate students all the way to researchers working in a broad spectrum of areas in analysis. Written with the graduate student in mind, it is assumed that the reader has familiarity with the basics of real analysis as well as with classical harmonic analysis, including Calderón-Zygmund theory; also some knowledge of basic orthogonal polynomials theory would be convenient. The monograph  develops the main topics of classical harmonic analysis (semigroups, covering lemmas, maximal functions, Littlewood-Paley functions, spectral multipliers, fractional integrals and fractional derivatives, singular integrals) with respect to the Gaussian measure. The text provide an updated exposition, as self-contained as possible, of all the topics in Gaussian harmonic analysis that up to now are mostly scattered in research papers and sections of books; also an exhaustive bibliography for further reading.  Each chapter ends with a section of notes and further results where connections between Gaussian harmonic analysis and other connected fields, points of view and alternative techniques are given. Mathematicians and researchers in several areas will find the breadth and depth of the treatment of the subject highly useful.

Martingale theory is a cornerstone of modern probability, offering a natural extension of the study of sums of independent random variables. Although its roots can be traced back to the work of Paul Lévy in 1937, it was Joseph L. Doob in the 1940s who formally developed the theory, culminating in his landmark book Stochastic Processes in 1953. Since then, martingale theory has evolved significantly, with deep contributions from mathematicians such as Donald L. Burkholder, Richard Gundy, and Burgess Davis, among others. This is what is now known as advanced martingale theory, which began with the publication of Burkholder’s seminal paper Martingale Transforms in 1966.This book provides a comprehensive treatment of both classical and advanced martingale theory. It opens with a historical introduction, exploring foundational functions such as Rademacher, Haar, and Walsh functions, before delving into the core concepts of conditional probability. The classical theory, as developed by Doob, is meticulously presented, followed by an in-depth examination of modern advancements, including Burkholder’s inequalities, Burkholder-Davis-Gundy inequality, and their generalizations, as well as good-lambda inequalities. The final chapter showcases a wide range of applications, highlighting the theory’s profound impact on Banach space theory, harmonic analysis, and beyond.Intended for graduate students and researchers in probability and analysis, this book serves as both an introduction and a reference, offering a clear and structured approach to a subject that continues to shape mathematical research and its applications.