A note on vector-valued Hardy and Paley inequalities

Abstract

The values of p and q for Lp(Lq) that satisfy the extension of Paley and Hardy inequalities for vector-valued HI functions are characterized. In particular, it is shown that L2(L1) is a Paley space that fails Hardy inequality. INTRODUCTION In [BP] the vector-valued analogue of two classical inequalities in the theory of Hardy spaces were investigated. A complex Banach space X is said to be a Paley space if /00 1/2 (P) (E IIf(2k)II2 ? CIfflI for all f E H1(X). k=0 A complex Banach space X is said to verify vector-valued Hardy inequality (for short X is a (HI)-space) if (H) IIf(n)I' < Cllfl II for all f e H1 (X), n=O where H1(X) = {f E L1(T, X): f(n) = O for n < O}. Both inequalities can be regarded in the framework of vector-valued extensions of multipliers from H1 to II . Recall that a sequence (mn) is a (H1 -1')multiplier, to be denoted by mn E (H1 /1), if Tmn (f) = (f(n)mO) defines a bounded operator from H1 into II . The (H1 I1)-multipliers were characterized by C. Fefferman in the following way (see [SW] for a proof):