Multi-parameter estimates via operator-valued shifts

We prove the mixed-norm L^pL^q -boundedness of a general class of singular integral operators having a multi-parameter singularity and acting on vector-valued (UMD Banach lattice-valued) functions. Moreover, families of such operators with uniform assumptions are shown to be not only uniformly bounded but R-bounded, a genuinely stronger property that is often needed in applications. Previous results of this nature only dealt with convolution-type or slightly more general paraproduct-free singular integrals. In contrast, our analysis specifically targets the array of different partial paraproducts that arise in the multi-parameter setting by interpreting them as paraproduct-valued one-parameter operators. This new point-of-view provides a conceptual simplification over the existing representation results for multi-parameter operators, which is a key to the proof of the boundedness of these operators.