Pseudodifferential analysis on conformally compact spaces
Author(s)
Bibliographic Information
Pseudodifferential analysis on conformally compact spaces
（Memoirs of the American Mathematical Society, no. 777）
American Mathematical Society, 2003
Available at 14 libraries
Note
"May 2003, volume 163, number 777 (fourth of 5 numbers)"
Includes bibliographical references (p. 79-83) and index
Description and Table of Contents
Description
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 Frechet $*$-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.
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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