Bounded $H_\infty $-calculus for pseudodifferential operators and applications to the Dirichlet-Neumann operator
Joachim Escher, Jörg Seiler · Transactions of the American Mathematical Society · 2008
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.