The sectorial projection defined from logarithms.

Gerd Grubb

The sectorial projection defined from logarithms.

Department of Mathematical Sciences

4 Citations (Scopus)

Abstract

For a classical elliptic pseudodifferential operator P of order m > 0 on a closed manifold X, such that the eigenvalues of the principal symbol p _m(x) have arguments in [θ, φ] and [φ, θ + 2π] (θ < φ < θ + 2 π), the sectorial projection Π_θ,φ(P) is defined essentially as the integral of the resolvent along e^iφR̄+ ∪ ei^θR+. In a recent paper, Booss-Bavnbek, Chen, Lesch and Zhu have pointed out that there is a flaw in several published proofs that Π_θ,φ(P) is a Ψ do of order 0; namely that p_m(x, ξ) cannot in general be modified to allow integration of (p_m(x, ξ) - λ) ^-1 along e^iφR̄₊ ∪ ei ^θR+ simultaneously for all ξ. We show that the structure of π_θ,φ(P) as a Ψ do of order 0 can be deduced from the formula Π_θ,φ(P) = i/2π(log_θ P - log_φ P) proved in an earlier work (coauthored with Gaarde). In the analysis of log_θ P one need only modify p_m(x, ξ) in a neighborhood of e^iθ R̄₊; this is known to be possible from Seeley's 1967 work on complex powers.

Translated title of the contribution	Den sektorielle projektion defineret fra logaritmer
Original language	English
Journal	Mathematica Scandinavica
Volume	111
Pages (from-to)	118-126
Number of pages	9
ISSN	0025-5521
Publication status	Published - 2012

Keywords

Faculty of Science

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title = "The sectorial projection defined from logarithms.",

abstract = "For a classical elliptic pseudodifferential operator P of order m > 0 on a closed manifold X, such that the eigenvalues of the principal symbol p m(x) have arguments in [θ, φ] and [φ, θ + 2π] (θ < φ < θ + 2 π), the sectorial projection Πθ,φ(P) is defined essentially as the integral of the resolvent along eiφ{\=R}+ ∪ eiθR+. In a recent paper, Booss-Bavnbek, Chen, Lesch and Zhu have pointed out that there is a flaw in several published proofs that Πθ,φ(P) is a Ψ do of order 0; namely that pm(x, ξ) cannot in general be modified to allow integration of (pm(x, ξ) - λ) -1 along eiφ{\=R}+ ∪ ei θR+ simultaneously for all ξ. We show that the structure of πθ,φ(P) as a Ψ do of order 0 can be deduced from the formula Πθ,φ(P) = i/2π(logθ P - logφ P) proved in an earlier work (coauthored with Gaarde). In the analysis of logθ P one need only modify pm(x, ξ) in a neighborhood of eiθ {\=R}+; this is known to be possible from Seeley's 1967 work on complex powers.",

keywords = "Faculty of Science, Matematik, partielle differentialligninger, geometrisk analyse",

author = "Gerd Grubb",

year = "2012",

language = "English",

volume = "111",

pages = "118--126",

journal = "Mathematica Scandinavica",

issn = "0025-5521",

publisher = "Aarhus Universitet * Mathematica Scandinavica",

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TY - JOUR

T1 - The sectorial projection defined from logarithms.

AU - Grubb, Gerd

PY - 2012

Y1 - 2012

N2 - For a classical elliptic pseudodifferential operator P of order m > 0 on a closed manifold X, such that the eigenvalues of the principal symbol p m(x) have arguments in [θ, φ] and [φ, θ + 2π] (θ < φ < θ + 2 π), the sectorial projection Πθ,φ(P) is defined essentially as the integral of the resolvent along eiφR̄+ ∪ eiθR+. In a recent paper, Booss-Bavnbek, Chen, Lesch and Zhu have pointed out that there is a flaw in several published proofs that Πθ,φ(P) is a Ψ do of order 0; namely that pm(x, ξ) cannot in general be modified to allow integration of (pm(x, ξ) - λ) -1 along eiφR̄+ ∪ ei θR+ simultaneously for all ξ. We show that the structure of πθ,φ(P) as a Ψ do of order 0 can be deduced from the formula Πθ,φ(P) = i/2π(logθ P - logφ P) proved in an earlier work (coauthored with Gaarde). In the analysis of logθ P one need only modify pm(x, ξ) in a neighborhood of eiθ R̄+; this is known to be possible from Seeley's 1967 work on complex powers.

AB - For a classical elliptic pseudodifferential operator P of order m > 0 on a closed manifold X, such that the eigenvalues of the principal symbol p m(x) have arguments in [θ, φ] and [φ, θ + 2π] (θ < φ < θ + 2 π), the sectorial projection Πθ,φ(P) is defined essentially as the integral of the resolvent along eiφR̄+ ∪ eiθR+. In a recent paper, Booss-Bavnbek, Chen, Lesch and Zhu have pointed out that there is a flaw in several published proofs that Πθ,φ(P) is a Ψ do of order 0; namely that pm(x, ξ) cannot in general be modified to allow integration of (pm(x, ξ) - λ) -1 along eiφR̄+ ∪ ei θR+ simultaneously for all ξ. We show that the structure of πθ,φ(P) as a Ψ do of order 0 can be deduced from the formula Πθ,φ(P) = i/2π(logθ P - logφ P) proved in an earlier work (coauthored with Gaarde). In the analysis of logθ P one need only modify pm(x, ξ) in a neighborhood of eiθ R̄+; this is known to be possible from Seeley's 1967 work on complex powers.

KW - Faculty of Science

KW - Matematik

KW - partielle differentialligninger

KW - geometrisk analyse

M3 - Journal article

SN - 0025-5521

VL - 111

SP - 118

EP - 126

JO - Mathematica Scandinavica

JF - Mathematica Scandinavica

ER -

The sectorial projection defined from logarithms.

Abstract

Keywords

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