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Some remarks on Banach spaces in which martingale difference sequences are unconditional. (English) Zbl 0533.46008
It is proved that the boundness of the vector-valued Hilbert-transform on the circle \({\mathbb{T}}\), \(H\otimes Id_ X\) on the space \(L^ p\!_ X({\mathbb{T}})\) implies that the Banach space X has the so called UMD- property, which means that X-valued martingale difference sequences are unconditional in \(L^ p\!_ X (1<p<\infty)\). This result, providing the converse of a theorem due to D. Burkholder ad T. McConnell, is already of interest in the scalar case, since it deduces Paley’s theorem from the M. Riesz theorem on conjugate functions. In the second part of the paper, the problem whether or not superreflexive lattices need to have the UMD- property is solved (by constructing a counterexample).

MSC:
46B42 Banach lattices
46B10 Duality and reflexivity in normed linear and Banach spaces
60G46 Martingales and classical analysis
46E40 Spaces of vector- and operator-valued functions
46B15 Summability and bases; functional analytic aspects of frames in Banach and Hilbert spaces
42A50 Conjugate functions, conjugate series, singular integrals
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