Hardy Martingales

This book presents the probabilistic methods around Hardy martingales for an audience interested in applications to complex, harmonic, and functional analysis. Building on work of Bourgain, Garling, Maurey, Pisier, and Varopoulos, it discusses in detail those martingale spaces that reflect characteristic qualities of complex analytic functions. Its particular themes are holomorphic random variables on Wiener space, and Hardy martingales on the infinite torus product, and their numerous deep applications to the geometry and classification of complex Banach spaces, e.g. the embedding of L1 into L1/H1, the isomorphic classification theorem for the class of poly-disk algebras, or the real variables characterization of Banach spaces with the analytic Radon Nikodym property. Including key background material on stochastic analysis and Banach space theory, it is suitable for a wide spectrum of researchers and graduate students working in classical and functional analysis.