We prove spectral multiplier theorems for Hörmander classes H α p for 0-sectorial operators A on Banach spaces assuming a bounded H ∞ (Σ σ ) calculus for some σ ∈ (0, π) and norm and certain R-bounds on one of the following families of operators: the semigroup e −zA on C + , the wave operators e isA for s ∈ R, the resolvent (λ − A) −1 on C\R, the imaginary powers A it for t ∈ R or the Bochner-Riesz means (1 − A/u) α + for u > 0. In contrast to the existing literature we neither assume that A operates on an L p 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) H α 1 calculus in terms of R-boundedness of Bochner-Riesz means.

Abstract. We extend the recent results concerning boundedness of the maximal regularity operator on tent spaces. This leads us to develop a singular integral operator theory on tent spaces. Such operators have operator-valued kernels. A seemingly appropriate condition on the kernel is time-space decay measured by off-diagonal estimates with various exponents.