Abstract

We consider multiparameter singular integrals and pseudodifferential operators acting on mixed-norm Bochner spaces L p 1 , … , p N ( R n 1 × ⋯ × R n N ; X ) where X is a UMD Banach space satisfying Pisier's property ( α ) . These geometric conditions are shown to be necessary. We obtain a vector-valued version of a result by R. Fefferman and Stein, also providing a new, inductive proof of the original scalar-valued theorem. Then we extend a result of Bourgain on singular integrals in UMD spaces with an unconditional basis to a multiparameter situation. Finally we carry over a result of Yamazaki on pseudodifferential operators to the Bochner space setting, improving the known vector-valued results even in the one-parameter case.

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