Bounded 𝐻_{∞}-calculus for pseudodifferential operators and applications to the Dirichlet-Neumann operator

Escher, J.; Seiler, J.

doi:10.1090/S0002-9947-08-04589-3

Bounded $H_\infty$-calculus for pseudodifferential operators and applications to the Dirichlet-Neumann operator
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by J. Escher and J. Seiler PDF

Trans. Amer. Math. Soc. 360 (2008), 3945-3973 Request permission

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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