TY - JOUR

T1 - Complex powers and non-compact manifolds

AU - Ammann, Bernd

AU - Lauter, Robert

AU - Nistor, Victor

AU - Vasy, András

N1 - Funding Information:
V. N. was partially supported by NSF Grants DMS 99-1981 and DMS 02-00808. A. V. was partially supported by NSF Grant DMS 02-01092 and by a Fellowship from the Sloan Foundation. B. A. was partially supported by The European Contract Human Potential Programme, Research Training Networks HPRN-CT-2000-00101 and HPRN-CT-1999-00118.

PY - 2004

Y1 - 2004

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]).

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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 -