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.

“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)

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).