Some remarks on Banach spaces in which martingale difference sequences are unconditional
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).
boundness of the vector-valued Hilbert-transform
scientific article; zbMATH DE number 3846212