Martingales and singular integrals in Banach spaces (Q2760172)

This paper surveys results on the interaction between Banach spaces and martingales. The results covered include the characterization of UMD spaces by the geometric concept of \(\zeta\)-convexity and with the help of the Hilbert transform and other singular integral operators, the precise value of the unconditionality constant of the Haar basis in \(L^p(0,1)\), the characterization of Hilbert spaces by the uniform boundedness of differentially subordinate martingales (giving best constants on the way), the complex unconditionality constant of the Haar basis in \(L^p_C(0,1)\), inequalities for the martingale square function, best estimates for differentially subordinate harmonic functions, applications to the Beurling-Ahlfors transform, and a characterization of spaces with the Radon-Nikodým property (RNP) using martingales. Then the paper briefly discusses the analytic RNP and gives, without proof, an analogous characterization of spaces having this property using analytic martingales. Finally, analytic UMD spaces are discussed and several characterizations of these spaces are mentioned. The paper finishes with an extensive list of references. All results are given with complete proofs and open problems are mentioned throughout the paper.NEWLINENEWLINEFor the entire collection see [Zbl 0970.46001].