# Pseudodifferential analysis on conformally compact spaces

### About this Title

**Robert Lauter**

Publication: Memoirs of the American Mathematical Society

Publication Year
2003: Volume 163, Number 777

ISBNs: 978-0-8218-3272-1 (print); 978-1-4704-0375-1 (online)

DOI: http://dx.doi.org/10.1090/memo/0777

MathSciNet review: 1965451

MSC: Primary 58J40; Secondary 35S05, 46L45, 47G30, 47L80, 58J05

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The $0$-calculus on a manifold with boundary is a
micro-localization of the Lie algebra of vector fields that vanish at the
boundary. It has been used by Mazzeo, Melrose to study the Laplacian of a
conformally compact metric. We give a complete characterization of those
$0$-pseudodifferential operators that are Fredholm between appropriate
weighted Sobolev spaces, and describe $C^{*}$-algebras that are
generated by $0$-pseudodifferential operators. An important step is
understanding the so-called reduced normal operator, or, almost equivalently,
the infinite dimensional irreducible representations of
$0$-pseudodifferential operators. Since the $0$-calculus
itself is not closed under holomorphic functional calculus, we construct
submultiplicative Fréchet $*$-algebras that contain and share
many properties with the $0$-calculus, and are stable under
holomorphic functional calculus ($\Psi^{*}$-algebras in the sense of
Gramsch). There are relations to elliptic boundary value problems.

Readership

Graduate students and research mathematicians interested in
analysis.

### Table of Contents

**Chapters**

- Part 1. Fredholm theory for $0$-pseudodifferential operators
- 1. Review on basic objects of 0-geometry
- 2. The small 0-calculus and the 0-calculus with bounds
- 3. The $b$-$c$-calculus on an interval
- 4. The reduced normal operator
- 5. Weighted 0-Sobolev spaces
- 6. Fredholm theory for 0-pseudodifferential operators
- Part 2. Algebras of $0$-pseudodifferential operators of order $0$
- 7. $C$*-algebras of 0-pseudodifferential operators
- 8. $\Psi $*-algebras of 0-pseudodifferential operators