Vector–valued Hardy Inequalities and B–convexity

Inequalities of the form ∑∞ k=0 |f̂(mk)| k+1 ≤ C ‖f‖1 for all f ∈ H1, where {mk} 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 < ∞ and 0 < α < ∞ the space X = H(1, q, α) consisting of analytic functions on the unit disc such that ∫ 1 0 (1 − r)qα−1M q 1 (f, r)dr < ∞ happens to satisfy the previous inequality for vector valued functions in H1(X), defined as the space of X-valued Bochner integrable functions on the torus whose negative Fourier coefficients vanish, for the case {mk} = {2k} but not for {mk} = {ka} for any a ∈ N. Introduction. In this paper we shall deal with the vector-valued formulation of certain inequalities in the theory of Hardy spaces. The first one, due to G. H. Hardy ([Du], page 48), reads ∞ ∑ n=0 |f̂(n)| n + 1 ≤ C ‖f‖1 for all f ∈ H (H) where H = {f ∈ L(T) : f̂(n) = 0 for n < 0} and, as usual, T stands for the unit circle and f̂(n) = ∫ π −π f(t)e −int dt 2π for n ∈ Z. Recently K. M. Dyakonov [D] considered the following generalized Hardy inequality: There exists a constant C > 0 such that, for any increasing subsequence {nk} of N satisfying δ = inf k∈N k nk+1 − nk nk > 0 (∗) one has ∞ ∑ k=0 |f̂(nk)| k + 1 ≤ C(1 + 1 δ ) ‖f‖1 for all f ∈ H. (GH) 1980 Mathematics Subject Classification (1985 Revision). Primary 42A45, 46E40; Secondary 42B30, 46B20.