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Pseudodifferential Analysis on Conformally Compact Spaces
Robert Lauter, University of Mainz, Germany
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Memoirs of the American Mathematical Society
2003; 92 pp; softcover
Volume: 163
ISBN-10: 0-8218-3272-7
ISBN-13: 978-0-8218-3272-1
List Price: US$60 Individual Members: US$36
Institutional Members: US\$48
Order Code: MEMO/163/777

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.

Graduate students and research mathematicians interested in analysis.

Part 1. Fredholm theory for $$0$$-pseudodifferential operators
• Review of basic objects of $$0$$-geometry
• The small $$0$$-calculus and the $$0$$-calculus with bounds
• The $$b$$-$$c$$-calculus on an interval
• The reduced normal operator
• Weighted $$0$$-Sobolev spaces
• Fredholm theory for $$0$$-pseudodifferential operators
Part 2. Algebras of $$0$$-pseudodifferential operators of order $$0$$
• $$C^*$$-algebras of $$0$$-pseudodifferential operators
• $$\Psi^*$$-algebras of $$0$$-pseudodifferential operators
• Appendix A. Spaces of conormal functions
• Bibliography
• Notations
• Index