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Bounded $ H_\infty$-calculus for pseudodifferential operators and applications to the Dirichlet-Neumann operator


Authors: J. Escher and J. Seiler
Journal: Trans. Amer. Math. Soc. 360 (2008), 3945-3973
MSC (2000): Primary 47G30; Secondary 35R35, 47A60, 58D25
DOI: https://doi.org/10.1090/S0002-9947-08-04589-3
Published electronically: March 13, 2008
MathSciNet review: 2395160
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Abstract: Operators of the form $ A=a(x,D)+K$ with a pseudodifferential symbol $ a(x,\xi)$ belonging to the Hörmander class $ S^m_{1,\delta}$, $ m>0$, $ 0\le\delta<1$, and certain perturbations $ K$ are shown to possess a bounded $ H_\infty$-calculus in Besov-Triebel-Lizorkin and certain subspaces of Hölder spaces, provided $ a$ is suitably elliptic. Applications concern pseudodifferential operators with mildly regular symbols and operators on manifolds of low regularity. An example is the Dirichlet-Neumann operator for a compact domain with $ \mathcal{C}^{1+r}$-boundary.


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Additional Information

J. Escher
Affiliation: Institut für Angewandte Mathematik, Universität Hannover, Welfengarten 1, 30167 Hannover, Germany
Email: escher@ifam.uni-hannover.de

J. Seiler
Affiliation: Institut für Angewandte Mathematik, Universität Hannover, Welfengarten 1, 30167 Hannover, Germany
Email: seiler@ifam.uni-hannover.de

DOI: https://doi.org/10.1090/S0002-9947-08-04589-3
Keywords: Bounded $H_\infty $-calculus, Dirichlet-Neumann operator, pseudodifferential operators
Received by editor(s): November 17, 2005
Published electronically: March 13, 2008
Article copyright: © Copyright 2008 American Mathematical Society

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