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

TL;DR

This paper establishes spectral multiplier theorems for sectorial operators on Banach spaces using $H^$ calculus and R-bounds, extending previous results to more general settings without requiring self-adjointness or $L^p$ structure.

Contribution

It introduces a unified approach to spectral multiplier theorems using R-bounds, applicable to a broader class of operators on Banach spaces, without assuming self-adjointness or $L^p$ frameworks.

Findings

Spectral multiplier theorems are proved under R-bounds for various operator families.

The results do not require the operator to be on an $L^p$ scale or self-adjoint.

A characterization of the $^_1$ calculus via R-bounds of Bochner-Riesz means is provided.

Abstract

We prove spectral multiplier theorems for H\"ormander 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.