ON THE NECESSITY OF PROPERTY (α) FOR SOME VECTOR-VALUED MULTIPLIER THEOREMS

Abstract. The notion of R-bounded operator families has proven to be cru-cial to several aspects of vector-valued Harmonic Analysis and its applications to PDEs. Concrete and important R-bounded sets have been identified among operators acting on UMD Banach spaces, which in addition enjoy G. Pisier’s property (α). We prove the necessity of (α) for a number of conclusions of this kind, complementing analogous results of G. Lancien in the product-space theory of Fourier multipliers and H∞-calculus. 1. Introduction an