Symmetric cones, possibly disguised under non-linear changes of coordinates, are the building blocks of manifolds with edges, corners, or conical points of a very general nature. Besides being a canonical open set of some Euclidean space, a symmetric cone L has an intrinsic Riemannian structure of its own, turning it into a symmetric space. These two structures make it possible to define on L a pseudodifferential analysis (the Fuchs calculus). The considerable interest in pseudodifferential problems on manifolds with non-smooth boundaries makes the precise analyses presented in this book both interesting and important. Much of the material in this book has never been previously published.
The methods used throughout the text rely heavily on the use of tools from quantum mechanics, such as representation theory and coherent states. Classes of operators defined by their symbols are given intrinsic characterizations. Harmonic analysis is discussed via the automorphism group of the complex tube over L.
The basic definitions governing the Fuchs calculus are provided, and a thorough exposition of the fundamental facts concerning the geometry of symmetric cones is given. The relationship with Jordan algebras is outlined and the general theory is illustrated by numerous examples.
The book offers the reader the technical tools for proving the main properties of the Fuchs calculus, with an emphasis on using the non-Euclidean Riemannian structure of the underlying cone. The fundamental results of pseudodifferential analysis are presented. The authors also develop the relationship to complex analysis and group representation.
This book benefits researchers interested in analysis on non-smooth domains or anyone working in pseudodifferential analysis. People interested in the geometry or harmonic analysis of symmetric cones will find in this valuable reference a new range of applications of complex analysis on tube-type symmetric domains and of the theory of Jordan algebras.
Table of Contents
General Definition of the Fuchs Calculus
The Geometry of Symmetric Cones
The Covariance Group of the Fuchs Calculus
Geometric Differential Inequalities
Weights and Classes of Symbols
The Family of m-Symbols
From Symbols to Operators: The Main Estimate and Continuity
From Operators to Symbols: The Converse of the Main Estimate
A Beals Characterization of Operators of Classical Type
Action of Diffeomorphisms on Operators of Classical Type
The l-Weyl Calculus: Unbounded Realization
Contraction of the l-Weyl Calculus
The l-Weyl Calculus: Bounded Realization
Contraction of the l-Weyl Calculus (Bounded Realization)
Unterberger\, Andre; Upmeier\, Harald