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This book presents a consistent development of the Kohn-Nirenberg type global quantization theory in the setting of graded nilpotent Lie groups in terms of their representations. It contains a detailed exposition of related background topics on homogeneous Lie groups, nilpotent Lie groups, and the analysis of Rockland operators on graded Lie groups together with their associated Sobolev spaces. For the specific example of the Heisenberg group the theory is illustrated in detail. In addition, the book features a brief account of the corresponding quantization theory in the setting of compact Lie groups.
The monograph is the winner of the 2014 Ferran Sunyer i Balaguer Prize.
List of contents
Preface.- Introduction.- Notation and conventions.- 1 Preliminaries on Lie groups.- 2 Quantization on compact Lie groups.- 3 Homogeneous Lie groups.- 4 Rockland operators and Sobolev spaces.- 5 Quantization on graded Lie groups.- 6 Pseudo-differential operators on the Heisenberg group.- A Miscellaneous.- B Group C* and von Neumann algebras.- Schrödinger representations and Weyl quantization.- Explicit symbolic calculus on the Heisenberg group.- List of quantizations.- Bibliography.- Index.
About the author
Veronique Fischer is a Senior Lecturer in Analysis at the University of Bath.
Michael Ruzhansky is a Professor of Pure Mathematics at Imperial College London.
The research of this monograph was supported by the
EPSRC Grant EP/K039407/1 when Veronique Fischer was employed at
Imperial College London. It started when she was working at the
University of Padua. The work was also supported by the
Marie Curie FP7 project (PseudodiffOperatorS - 301599) and by
the Leverhulme Trust (grant RPG-2014-02).
Summary
This book presents a consistent development of the Kohn-Nirenberg type global quantization theory in the setting of graded nilpotent Lie groups in terms of their representations. It contains a detailed exposition of related background topics on homogeneous Lie groups, nilpotent Lie groups, and the analysis of Rockland operators on graded Lie groups together with their associated Sobolev spaces. For the specific example of the Heisenberg group the theory is illustrated in detail. In addition, the book features a brief account of the corresponding quantization theory in the setting of compact Lie groups.
The monograph is the winner of the 2014 Ferran Sunyer i Balaguer Prize.
Additional text
“The main topic of this prize-winning monograph is the development of a pseudo-differential calculus on homogeneous Lie groups–the nilpotent Lie group equipped with a family of dilations compatible with the group structure. … It is really surprising that in spite of its great length and complicated subject, this book is very accessible.”(Antoni Wawrzyńczyk, Mathematical Reviews, April, 2017)“We want to remark that the contents of the volume are extremely rich. Beside presenting the new theory in the graded nilpotent case, the authors offer a complete view of the calculus of pseudo-differential operators on groups giving detailed references to preceding contributions. Also, we note the big effort to provide a self-contained presentation, addressed to a large audience. This monograph was the winner of the 2016 Ferran Sunyer i Balanguer prize.” (Luigi Rodino, zbMATH 1347.22001, 2016)
Report
"The main topic of this prize-winning monograph is the development of a pseudo-differential calculus on homogeneous Lie groups-the nilpotent Lie group equipped with a family of dilations compatible with the group structure. ... It is really surprising that in spite of its great length and complicated subject, this book is very accessible."(Antoni Wawrzynczyk, Mathematical Reviews, April, 2017)
"We want to remark that the contents of the volume are extremely rich. Beside presenting the new theory in the graded nilpotent case, the authors offer a complete view of the calculus of pseudo-differential operators on groups giving detailed references to preceding contributions. Also, we note the big effort to provide a self-contained presentation, addressed to a large audience. This monograph was the winner of the 2016 Ferran Sunyer i Balanguer prize." (Luigi Rodino, zbMATH 1347.22001, 2016)
