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 -