Complex powers and non-compact manifolds

Bernd Ammann, Robert Lauter, Victor Nistor, András Vasy

Research output: Contribution to journalArticle

16 Citations (Scopus)

Abstract

We study the complex powers Az of an elliptic, strictly positive pseudodifferential operator A using an axiomatic method that combines the approaches of Guillemin and Seeley. In particular, we introduce a class of algebras, called "Guillemin algebras," whose definition was inspired by Guillemin [Guillemin, V. (1985). A new proof of Weyl's formula on the asymptotic distribution of eigenvalues. Adv. in Math. 55:131-160]. A Guillemin algebra can be thought of as an algebra of "abstract pseudodifferential operators." Most algebras of pseudodifferential operators belong to this class. Several results typical for algebras of pseudodifferential operators (asymptotic completeness, construction of Sobolev spaces, boundedness between appropriate Sobolev spaces,...) generalize to Guillemin algebras. Most important, this class of algebras provides a convenient framework to obtain precise estimates at infinity for Az, when A > 0 is elliptic and defined on a non-compact manifold, provided that a suitable ideal of regularizing operators is specified (a submultiplicative Ψ*-algebra). We shall use these results in a forthcoming paper to study pseudodifferential operators and Sobolev spaces on manifolds with a Lie structure at infinity (a certain class of non-compact manifolds that has emerged from Melrose's work on geometric scattering theory [Melrose, R. B. (1995). Geometric Scattering Theory. Stanford Lectures. Cambridge: Cambridge University Press]).

Original languageEnglish (US)
Pages (from-to)671-705
Number of pages35
JournalCommunications in Partial Differential Equations
Volume29
Issue number5-6
DOIs
StatePublished - Jan 1 2004

Fingerprint

Noncompact Manifold
Algebra
Pseudodifferential Operators
Mathematical operators
Sobolev spaces
Sobolev Spaces
Scattering Theory
Infinity
Scattering
Operator Space
Positive Operator
Strictly positive
Asymptotic distribution
Boundedness
Completeness
Eigenvalue
Generalise
Class

All Science Journal Classification (ASJC) codes

  • Analysis
  • Applied Mathematics

Cite this

Ammann, Bernd ; Lauter, Robert ; Nistor, Victor ; Vasy, András. / Complex powers and non-compact manifolds. In: Communications in Partial Differential Equations. 2004 ; Vol. 29, No. 5-6. pp. 671-705.
@article{3e024a777d374786a9783d96f506a643,
title = "Complex powers and non-compact manifolds",
abstract = "We study the complex powers Az of an elliptic, strictly positive pseudodifferential operator A using an axiomatic method that combines the approaches of Guillemin and Seeley. In particular, we introduce a class of algebras, called {"}Guillemin algebras,{"} whose definition was inspired by Guillemin [Guillemin, V. (1985). A new proof of Weyl's formula on the asymptotic distribution of eigenvalues. Adv. in Math. 55:131-160]. A Guillemin algebra can be thought of as an algebra of {"}abstract pseudodifferential operators.{"} Most algebras of pseudodifferential operators belong to this class. Several results typical for algebras of pseudodifferential operators (asymptotic completeness, construction of Sobolev spaces, boundedness between appropriate Sobolev spaces,...) generalize to Guillemin algebras. Most important, this class of algebras provides a convenient framework to obtain precise estimates at infinity for Az, when A > 0 is elliptic and defined on a non-compact manifold, provided that a suitable ideal of regularizing operators is specified (a submultiplicative Ψ*-algebra). We shall use these results in a forthcoming paper to study pseudodifferential operators and Sobolev spaces on manifolds with a Lie structure at infinity (a certain class of non-compact manifolds that has emerged from Melrose's work on geometric scattering theory [Melrose, R. B. (1995). Geometric Scattering Theory. Stanford Lectures. Cambridge: Cambridge University Press]).",
author = "Bernd Ammann and Robert Lauter and Victor Nistor and Andr{\'a}s Vasy",
year = "2004",
month = "1",
day = "1",
doi = "10.1081/PDE-120037329",
language = "English (US)",
volume = "29",
pages = "671--705",
journal = "Communications in Partial Differential Equations",
issn = "0360-5302",
publisher = "Taylor and Francis Ltd.",
number = "5-6",

}

Ammann, B, Lauter, R, Nistor, V & Vasy, A 2004, 'Complex powers and non-compact manifolds', Communications in Partial Differential Equations, vol. 29, no. 5-6, pp. 671-705. https://doi.org/10.1081/PDE-120037329

Complex powers and non-compact manifolds. / Ammann, Bernd; Lauter, Robert; Nistor, Victor; Vasy, András.

In: Communications in Partial Differential Equations, Vol. 29, No. 5-6, 01.01.2004, p. 671-705.

Research output: Contribution to journalArticle

TY - JOUR

T1 - Complex powers and non-compact manifolds

AU - Ammann, Bernd

AU - Lauter, Robert

AU - Nistor, Victor

AU - Vasy, András

PY - 2004/1/1

Y1 - 2004/1/1

N2 - We study the complex powers Az of an elliptic, strictly positive pseudodifferential operator A using an axiomatic method that combines the approaches of Guillemin and Seeley. In particular, we introduce a class of algebras, called "Guillemin algebras," whose definition was inspired by Guillemin [Guillemin, V. (1985). A new proof of Weyl's formula on the asymptotic distribution of eigenvalues. Adv. in Math. 55:131-160]. A Guillemin algebra can be thought of as an algebra of "abstract pseudodifferential operators." Most algebras of pseudodifferential operators belong to this class. Several results typical for algebras of pseudodifferential operators (asymptotic completeness, construction of Sobolev spaces, boundedness between appropriate Sobolev spaces,...) generalize to Guillemin algebras. Most important, this class of algebras provides a convenient framework to obtain precise estimates at infinity for Az, when A > 0 is elliptic and defined on a non-compact manifold, provided that a suitable ideal of regularizing operators is specified (a submultiplicative Ψ*-algebra). We shall use these results in a forthcoming paper to study pseudodifferential operators and Sobolev spaces on manifolds with a Lie structure at infinity (a certain class of non-compact manifolds that has emerged from Melrose's work on geometric scattering theory [Melrose, R. B. (1995). Geometric Scattering Theory. Stanford Lectures. Cambridge: Cambridge University Press]).

AB - We study the complex powers Az of an elliptic, strictly positive pseudodifferential operator A using an axiomatic method that combines the approaches of Guillemin and Seeley. In particular, we introduce a class of algebras, called "Guillemin algebras," whose definition was inspired by Guillemin [Guillemin, V. (1985). A new proof of Weyl's formula on the asymptotic distribution of eigenvalues. Adv. in Math. 55:131-160]. A Guillemin algebra can be thought of as an algebra of "abstract pseudodifferential operators." Most algebras of pseudodifferential operators belong to this class. Several results typical for algebras of pseudodifferential operators (asymptotic completeness, construction of Sobolev spaces, boundedness between appropriate Sobolev spaces,...) generalize to Guillemin algebras. Most important, this class of algebras provides a convenient framework to obtain precise estimates at infinity for Az, when A > 0 is elliptic and defined on a non-compact manifold, provided that a suitable ideal of regularizing operators is specified (a submultiplicative Ψ*-algebra). We shall use these results in a forthcoming paper to study pseudodifferential operators and Sobolev spaces on manifolds with a Lie structure at infinity (a certain class of non-compact manifolds that has emerged from Melrose's work on geometric scattering theory [Melrose, R. B. (1995). Geometric Scattering Theory. Stanford Lectures. Cambridge: Cambridge University Press]).

UR - http://www.scopus.com/inward/record.url?scp=2942657234&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=2942657234&partnerID=8YFLogxK

U2 - 10.1081/PDE-120037329

DO - 10.1081/PDE-120037329

M3 - Article

AN - SCOPUS:2942657234

VL - 29

SP - 671

EP - 705

JO - Communications in Partial Differential Equations

JF - Communications in Partial Differential Equations

SN - 0360-5302

IS - 5-6

ER -