Analysis in Banach Spaces: Volume III: Harmonic Analysis and Spectral Theory(76 Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge / A Series of Modern Surveys in Mathematics)

This third volume of Analysis in Banach Spaces offers a systematic treatment of Banach space-valued singular integrals, Fourier transforms, and function spaces. It further develops and ramifies the theory of functional calculus from Volume II and describes applications of these new notions and tools to the problem of maximal regularity of evolution equations.  The exposition provides a unified treatment of a large body of results, much of which has previously only been available in the form of research papers. Some of the more classical topics are presented in a novel way using modern techniques amenable to a vector-valued treatment. Thanks to its accessible style with complete and detailed proofs, this book will be an invaluable reference for researchers interested in functional analysis, harmonic analysis, and the operator-theoretic approach to deterministic and stochastic evolution equations.

Table of Contents:
11 Singular integral operators.- 12 Dyadic operators and the T (1) theorem.- 13 The Fourier transform and multipliers.- 14 Function spaces.- 15 Bounded imaginary powers.- 16 The H∞-functional calculus revisited.- 17  Maximal regularity.- 18 Nonlinear parabolic evolution equations in critical spaces.-  Appendix Q: Questions.-  Appendix K: Semigroup theory revisited.- Appendix L: The trace method for real interpolation theory.

About the Author :
Tuomas Hytönen is Professor at the University of Helsinki. A leading expert in Harmonic Analysis with over 100 research papers, he was educated at Helsinki University of Technology and spent a postdoc year at Delft University of Technology. He received a European Research Council Starting Grant in 2011 and gave an invited address to the International Congress of Mathematicians in 2014. Jan van Neerven is Professor of Analysis at Delft University of Technology. Author of more than 120 research papers and two monographs, he is a leading expert in Operator Theory and Stochastic Analysis. He held post-doctoral positions at Caltech and Tübingen University. He was awarded a Human Capital and Mobility fellowship, a fellowship of the Royal Dutch Academy of Arts and Sciences, and VIDI and VICI subsidies from the Netherlands Organisation for Scientific Research. Mark Veraar is Professor at Delft University of Technology. Author of over 80 research papers, he is a leading researcher in the theory of evolution equations and stochastic partial differential equations. He held post-doctoral positions at the Universities of Warsaw and Karlsruhe, the latter with a Alexander von Humboldt Fellowship. He is the recipient of VENI, VIDI, and VICI grants from the Netherlands Organisation for Scientific Research. Lutz Weis, a Professor Emeritus at Karlsruhe Institute of Technology, is a senior researcher in operator theory and evolution equations. He has published over 90 research papers and a monograph. Since receiving his PhD from University of Bonn, he was a professor at Louisiana State University and visiting professor at TU Berlin as well as Universities of Kiel, South Carolina and Minnesota. He organized a Marie Curie training site and is currently a member of a DFG Graduiertenkolleg.  

Review :
“The authors thus select a representative set of topics, and present each of them in the thorough and highly readable style … . Anyone primarily trained in functional analysis is extremely likely to enjoy the Analysis in Banach spaces series. For these readers, the series provides a highly readable and completely self-contained entry point into many other Fields, especially harmonic and stochastic analysis.” (Pierre Portal, Mathematical Reviews, January, 2025) “The authors always cover the necessary prerequisites from earlier developments and the book is meant to be self-contained. ... The volume ends with an interesting list of open problems and an appendix containing a section on measurable semigroups and another one on the trace method for real interpolation.” (Oscar Blasco, zbMATH 1534.46003, 2024)