Arkiv för Matematik

Vector-valued Hardy inequalities and B-convexity

Oscar Blasco

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Inequalities of the form $\sum\nolimits_{k = 0}^\infty {|\hat f(m_k )|/(k + 1) \leqslant C||f||_1 } $ for all fH1, 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 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 {mk}={2k} but not for {mk}={ka} for any α ∈ N.


The author has been partially supported by the Spanish DGICYT, Proyecto PB95-0291.

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Ark. Mat., Volume 38, Number 1 (2000), 21-36.

Received: 23 September 1998
First available in Project Euclid: 31 January 2017

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2000 © Institut Mittag-Leffler


Blasco, Oscar. Vector-valued Hardy inequalities and B -convexity. Ark. Mat. 38 (2000), no. 1, 21--36. doi:10.1007/BF02384487.

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