Theorems of Hardy and Paley for vector-valued analytic functions and related classes of Banach spaces

Abstract

We investigate the classes of Banach spaces where analogues of the classical Hardy inequality and the Paley gap theorem hold for vector-valued functions. We show that the vector-valued Paley theorem is valid for a large class of Banach spaces (necessarily of cotype 2 2 ) which includes all Banach lattices of cotype 2 2 , all Banach spaces whose dual is of type 2 2 and also the preduals of C ∗ {C^ * } -algebras. For the trace class S 1 {S_1} and the dual of the algebra of all bounded operators on a Hilbert space a stronger result holds; namely, the vector-valued analogue of the Fefferman theorem on multipliers from H 1 {H^1} into l 1 {l^1} ; in particular for the latter spaces the vector-valued Hardy inequality holds. This inequality is also true for every Banach space of type > 1 > 1 (Bourgain).