Preface

Over the past fifteen years, motivated by regularity problems in evolution equations, there has been tremendous progress in the analysis of Banach space-valued functions and processes. For so-called UMD spaces in particular, central areas of Harmonic Analysis, such as the theory of Fourier multipli-ers and singular integrals, were extended to operator-valued kernels acting on Bochner spaces, and crucial estimates of Stochastic Analysis, including the Ito ̂ isometry and the Burkholder–Davis–Gundy inequalities, were generalised to Banach space-valued processes. As it was long known that extensions of such sophisticated scalar-valued estimates are not possible for all Banach spaces, these results depended on es-sential progress in the Geometry of Banach spaces during the 70s and 80s. The theory of Burkholder and Bourgain on UMD spaces became the foundation on which the recent theory we wish to report on was built; just as important are results of Maurey, Pisier, and Kwapien ́ on type and cotype, since they link the structure of the Banach space to estimates for random sums whic