Aspects of probabilistic Littlewood-Paley theory in Banach spaces

摘要

A probabilistic square function estimate, due to Varopoulos in the scalar case, is proved for functions with values in a UMD Banach space. As an application, close to optimal dimensionfree norm bounds for certain spectral multipliers of the Laplacian in Bochner spaces are obtained. The proof uses stochastic integration of vector-valued processes and especially decoupling estimates due to Garling.