On the necessity of property (alpha) for some vector-valued multiplier theorems

摘要

The notion of R-bounded operator families has proven to be crucial 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 (alpha). We prove the necessity of (alpha) for a number of conclusions of this kind, complementing analogous results of G. Lancien in the product-space theory of Fourier multipliers and H-infinity-calculus.