## Arkiv för Matematik

### Vector-valued Hardy inequalities and B-convexity

Oscar Blasco

#### Abstract

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.

#### Note

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

#### Article information

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

Dates
First available in Project Euclid: 31 January 2017

https://projecteuclid.org/euclid.afm/1485898663

Digital Object Identifier
doi:10.1007/BF02384487

Mathematical Reviews number (MathSciNet)
MR1749355

Zentralblatt MATH identifier
1028.42016

Rights

#### Citation

Blasco, Oscar. Vector-valued Hardy inequalities and B -convexity. Ark. Mat. 38 (2000), no. 1, 21--36. doi:10.1007/BF02384487. https://projecteuclid.org/euclid.afm/1485898663

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