Spectral multiplier theorems via H-infinity calculus and R-bounds

摘要

We prove spectral multiplier theorems for Hormander classes for 0-sectorial operators A on Banach spaces assuming a bounded calculus for some and norm and certain R-bounds on one of the following families of operators: the semigroup on the wave operators for the resolvent on the imaginary powers for or the Bochner-Riesz means for In contrast to the existing literature we neither assume that A operates on an scale nor that A is self-adjoint on a Hilbert space. Furthermore, we replace (generalized) Gaussian or Poisson bounds and maximal estimates by the weaker notion of R-bounds, which allow for a unified approach to spectral multiplier theorems in a more general setting. In this setting our results are close to being optimal. Moreover, we can give a characterization of the (R-bounded) calculus in terms of R-boundedness of Bochner-Riesz means.