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Transactions of the American Mathematical Society

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Theorems of Hardy and Paley for vector-valued analytic functions and related classes of Banach spaces

Authors: O. Blasco and A. Pełczyński
Journal: Trans. Amer. Math. Soc. 323 (1991), 335-367
MSC: Primary 46B20; Secondary 42B30, 46E15, 46L99, 47D15
MathSciNet review: 979957
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Abstract: We investigate the classes of Banach spaces where analogues of the classical Hardy inequality and the Paley gap theorem hold for vector-valued functions. We show that the vector-valued Paley theorem is valid for a large class of Banach spaces (necessarily of cotype $ 2$) which includes all Banach lattices of cotype $ 2$, all Banach spaces whose dual is of type $ 2$ and also the preduals of $ {C^ * }$-algebras. For the trace class $ {S_1}$ and the dual of the algebra of all bounded operators on a Hilbert space a stronger result holds; namely, the vector-valued analogue of the Fefferman theorem on multipliers from $ {H^1}$ into $ {l^1}$; in particular for the latter spaces the vector-valued Hardy inequality holds. This inequality is also true for every Banach space of type $ > 1$ (Bourgain).

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Keywords: Hardy inequality, Paley theorem, multipliers, $ X$-atoms, Banach spaces of type $ > 1$, cotype $ 2$
Article copyright: © Copyright 1991 American Mathematical Society

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