Multi‐parameter estimates via operator‐valued shifts

Abstract

We prove the mixed-norm L p L 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.

Off-diagonal estimates for commutators of bi-parameter singular integrals

“…The following Proposition 5.4 is e.g. contained as a part of Hytönen-Martikainen-Vuorinen [21]. 5.4.…”

Off-diagonal estimates for bi-commutators

“…Instead of dealing with UMD spaces with property (α) we will be working with the formally stronger (although for concrete examples essentially the same) assumption that X is a UMD function lattice. For proofs and background on function lattices see [1,23,36,37]. A normed space X is a Banach function lattice if the following four conditions hold.…”

“…and define S D 2 ,M D 1 f analogously. We are ready to state the following standard estimates that are at least recorded in [23].…”

Exotic Calderón–Zygmund Operators

“…For recent work using this viewpoint see e.g. [11]. Developing such an approach to Zygmund SIOs is interesting.…”