Spectral multiplier theorems via $H^{\infty }$ calculus and $R$-bounds

Abstract

We prove spectral multiplier theorems for Hörmander classes ${\mathcal{H}}^{\alpha }\_p$ for 0-sectorial operators A on Banach spaces assuming a bounded $H^{\infty }(\Sigma \_\sigma )$ calculus for some $\sigma \in (0,\pi )$ and norm and certain R-bounds on one of the following families of operators: the semigroup $e^{--zA}$ on $\mathbb{C}\_+$, the wave operators $e^{isA}$ for $s\in \mathbb{R}$, the resolvent $(\lambda --A{)}^{-1}$ on $\mathbb{C}\backslash \mathbb{R}$, the imaginary powers $A^{it}$ for $t\in \mathbb{R}$ or the Bochner-Riesz means $(1-A/u{)}^{\alpha }\_+$ for $u>0.$ In contrast to the existing literature we neither assume that A operates on an Lp 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) ${\mathcal{H}}^{\alpha }\_1$ calculus in terms of R-boundedness of Bochner-Riesz means.

Mathematische Zeitschrift, Springer, 2018

References in corpus (3)

• Paley-Littlewood decomposition for sectorial operators and interpolation spaces
• The $H^{\infty }$-Functional Calculus and Square Function Estimates
• The Kalton-Lancien Theorem Revisited: Maximal Regularity does not extrapolate