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

Blasco, O.; Pełczyński, A.

doi:10.1090/S0002-9947-1991-0979957-5

Theorems of Hardy and Paley for vector-valued analytic functions and related classes of Banach spaces
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by O. Blasco and A. Pełczyński PDF

Trans. Amer. Math. Soc. 323 (1991), 335-367 Request permission

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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