Riesz Transforms, Hodge-Dirac Operators and Functional Calculus for Multipliers

The book discusses recent research in noncommutative harmonic analysis, focusing on the L_p boundedness of Riesz transforms associated with Markovian semigroups of Fourier or Schur multipliers on non-abelian groups. It also presents a proof of the boundedness of the holomorphic functional calculus for Hodge-Dirac operators and discusses the connection with noncommutative geometry. The theory builds on the continuity of the Hilbert and Riesz transforms on L_p and provides a self-contained introduction to the requisite noncommutative background.

Format: Paperback / softback
Length: 280 pages
Publication date: 06 May 2022
Publisher: Springer Nature Switzerland AG

This book delves into the realm of recent research in noncommutative harmonic analysis, focusing on the study of L^p boundedness for Riesz transforms associated with Markovian semigroups. These semigroups can be characterized by either Fourier multipliers on non-abelian groups or Schur multipliers, and the book provides a comprehensive exploration of these objects.

The detailed study of these Riesz transforms is carried forward with a proof of the boundedness of the holomorphic functional calculus for Hodge-Dirac operators. This groundbreaking result answers a question posed by Junge, Mei, and Parcet and presents a novel functional analytic approach that opens up new avenues for exploring the connection between noncommutative geometry and these L^p operations.

Furthermore, the book showcases how these L^p operations yield new examples of quantum compact metric spaces and spectral triples. The theoretical framework underpinning this book is rooted in one of the twentieth century's most significant discoveries in analysis: the continuity of the Hilbert and Riesz transforms on L^p. In the works of Lust-Piquard (1998) and Junge, Mei, and Parcet (2018), it became evident that these L^p operations can be formulated on L^p spaces associated with groups.

Continuing this line of research, the book offers a self-contained introduction to the necessary noncommutative background. Covering an active and exciting topic with numerous connections to recent developments in noncommutative harmonic analysis, this book will appeal to experts in noncommutative L^p spaces as well as analysts interested in constructing Riesz transforms and Hodge-Dirac operators.

Weight: 450g
Dimension: 235 x 155 (mm)
ISBN-13: 9783030990107
Edition number: 1st ed. 2022