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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$57
Individual Members: US$34.20
Institutional Members: US$45.60
Order Code: MEMO/163/777
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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

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
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