Modulation Spaces

This monograph serves as a much-needed, self-contained reference on the topic of modulation spaces. By gathering together state-of-the-art developments and previously unexplored applications, readers will be motivated to make effective use of this topic in future research. Because modulation spaces have historically only received a cursory treatment, this book will fill a gap in time-frequency analysis literature, and offer readers a convenient and timely resource.

Foundational concepts and definitions in functional, harmonic, and real analysis are reviewed in the first chapter, which is then followed by introducing modulation spaces. The focus then expands to the many valuable applications of modulation spaces, such as linear and multilinear pseudodifferential operators, and dispersive partial differential equations. Because it is almost entirely self-contained, these insights will be accessible to a wide audience of interested readers.

Modulation Spaces will bean ideal reference for researchers in time-frequency analysis and nonlinear partial differential equations. It will also appeal to graduate students and seasoned researchers who seek an introduction to the time-frequency analysis of nonlinear dispersive partial differential equations.

Explores recent developments in the field of time-frequency analysis, particularly focusing on the topic of modulation spaces Presents valuable applications of modulation spaces to pseudodifferential operators and partial differential equations Appeals to a wide audience with its clear and self-contained presentation

Résumé

"The goal of this book is to provide an exhaustive treatment of the theory of modulation spaces. Together with detailed supplementary references, this book is particularly suitable for graduate students who want to apply modulation spaces to PDEs. ... this textbook can be recommended to the people who want to know what modulation spaces are." (Yoshihiro Sawano, Mathematical Reviews, March, 2023)

"The book contains a comprehensive theory of modulation spaces and applications to pseudo-differential operators and nonlinear Schrödinger equations. ... The book concentrates on the latest developments of the modulation spaces ... . The book is rather self-contained and of good exposition, and is a fine monograph ... ." (Renjin Jiang, zbMATH 1476.35001, 2022)

Contenu

Notions of real, functional and Fourier analysis.- Modulation spaces.- Equivalent definitions of modulation spaces.- Pseudodifferential operators.- Weighted modulation spaces.- Modulation spaces and other function spaces.- Applications to partial differential equations.- A proof of Banach's fixed point theorem.- The Picard-Lindelöf and Peano theorems.- Gronwall's lemma.- Local well-posedness of NLS on Sobolev spaces.