Vector-valued Hardy inequalities andB-convexity

Inequalities of the form for allf∈H ¹, where {m _k } are special subsequences of natural numbers, are investigated in the vector-valued setting. It is proved that Hardy's inequality and the generalized Hardy inequality are equivalent for vector valued Hardy spaces defined in terms ff atoms and that they actually characterizeB-convexity. It is also shown that for 1<q<∞ and 0<α<∞ the spaceX=H(1,q,γa) consisting of analytic functions on the unit disc such that satisfies the previous inequality for vector valued functions inH ¹ (X), defined as the space ofX-valued Bochner integrable functions on the torus whose negative Fourier coefficients vanish, for the case {m _k }={2^k} but not for {m _k }={k ^a } for any α ∈ N. The author has been partially supported by the Spanish DGICYT, Proyecto PB95-0291.