Quantization on Nilpotent Lie Groups

1. Front Matter

   Pages i-xxviii

   PDF

2. Preliminaries on Lie groups

   • Veronique Fischer, Michael Ruzhansky

   Pages 15-56Open Access

3. Quantization on compact Lie groups

   • Veronique Fischer, Michael Ruzhansky

   Pages 57-90Open Access

4. Homogeneous Lie groups

   • Veronique Fischer, Michael Ruzhansky

   Pages 91-170Open Access

5. Rockland operators and Sobolev spaces

   • Veronique Fischer, Michael Ruzhansky

   Pages 171-269Open Access

6. Quantization on graded Lie groups

   • Veronique Fischer, Michael Ruzhansky

   Pages 271-426Open Access

7. Pseudo-differential operators on the Heisenberg group

   • Veronique Fischer, Michael Ruzhansky

   Pages 427-489Open Access

8. Back Matter

   Pages 491-557

   PDF

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.

• graded Lie groups
• Heisenberg group
• compact Lie groups
• pseudo-differential operators
• Sobolev spaces
• singular integral operators

“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)

• Department of Math, University of Bath, Bath, United Kingdom

  Veronique Fischer

• Department of Mathematics, Imperial College London, London, United Kingdom

  Michael Ruzhansky

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).