Vector-valued Hardy inequalities andB-convexity
AbstractInequalities of the form
$$\sum\nolimits_{k = 0}^\infty  {|\hat f(m_k )|/(k + 1) \leqslant C||f||_1 } $$
 for allf∈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 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
$$\int_0^1 {(1 - r)^{q\alpha  - 1} M_1^q (f,r) dr<  \infty } $$
 satisfies the previous inequality for vector valued functions inH1 (X), defined as the space ofX-valued Bochner integrable functions on the torus whose negative Fourier coefficients vanish, for the case {mk}={2k} but not for {mk}={ka} for any α ∈ N.