Vector-valued Hardy inequalities and B-convexity

Inequalities of the form Sigma(k=0)(infinity)\(f) over cap (m(k))\/(k+1)less than or equal toCparallel tofparallel to(1) for all f is an element of 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 of atoms; and that they actually characterize B-convexity. It is also shown that for 1<q<infinity and 0<alpha<infinity the space X=H(1,q,alpha) consisting of analytic functions on the unit disc such that f(0)(1)(1-r)q(alpha-1) M-1(q)(f, r)dr<infinity satisfies the previous inequality for vector valued functions in H-1(X), defined as the space of X-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(alpha)} for any a is an element of N.