Complex powers on noncompact manifolds and manifolds with singularities
type: Item
instance of: scholarly article
title: Complex powers on noncompact manifolds and manifolds with singularities
zbMATH Open document ID: 0627.35090
DOI: 10.1007/BF01457152
author: Elmar Schrohe
published in: Mathematische Annalen
publication date: 1988-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/164428
review text: For an elliptic pseudodifferential operator (pdo) A of positive order a family of complex powers \(\{A^ s:\) \(s\in {\mathbb{C}}\}\) is constructed. The kernel function \(k_ s(x,y)\) of \(A^ s\) is a continuous function of (x,y) depending analytically on s for s in a half-plane of \({\mathbb{C}}\). Moreover, in the most interesting case \(x=y\), \(s\mapsto k_ s(x,x)\) is shown to have an analytic extension to a larger half-plane, the domain and the type of possible singularities depending on the asymptotic expansion of the symbol of A. In particular, \(s\mapsto k_ s(x,x)\) can be continued holomorphically into the origin \(s=0\) under corresponding assumptions. Similar results hold for the zeta and eta function of A. The proof uses the calculus of pdo's on SG-compatible manifolds (including manifolds with singularities) based on H. O. Cordes' idea of globally defined symbols and the classical techniques in complex powers of R. Seeley and H. Kumnao-go.
Mathematics Subject Classification ID: 35S05
Mathematics Subject Classification ID: 47Gxx
Mathematics Subject Classification ID: 58J40
zbMATH DE Number: 4019636
zbMATH Keywords: elliptic pseudodifferential operator
zbMATH Keywords: positive order
zbMATH Keywords: family of complex powers
zbMATH Keywords: analytic extension
zbMATH Keywords: singularities
zbMATH Keywords: asymptotic expansion of the symbol
zbMATH Keywords: SG-compatible manifolds
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 4019636