Vector-valued Hardy inequalities and B-convexity

Abstract

Inequalities of the form $\sum\nolimits_{k = 0}^\infty {|\hat f(m_k )|/(k + 1) \leqslant C||f||_1 } $ for all f∈H¹, 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 characterize B-convexity. It is also shown that for 1< q<∞ and 0<α<∞ the space X=H(1, q,γa) consisting of analytic functions on the unit disc such that $\int_0^1 {(1 - r)^{q\alpha - 1} M_1^q (f,r) dr< \infty } $ satisfies 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 {m_k}={2^k} but not for {m_k}={k^a} for any α ∈ N.