Product space singular integrals with mild kernel regularity
\textit{T. P. Hytönen} [Ann. Math. (2) 175, No. 3, 1473--1506 (2012; Zbl 1250.42036)] proved a stunning representation theorem that expresses any Calderón-Zygmund Singular Integral Operator (SIO) as a sum of dyadic shifts (basic operators that have a simple form with respect to Haar bases). Such a result allows one to efficiently prove sharp weighted \(L^p\) bounds for SIO by proving sharp estimates for dyadic shifts. This allowed Hytönen, for instance, to settle the \(A^2\) conjecture in [loc. cit.]. This representation theorem has since been extended to cover more general classes of SIO, and the paper under review can be seen as a culmination of this line of research. It treats SIO that have rougher kernels than traditional Calderón-Zygmund operators, more general singularities (when \(x_{1}=y_{1}\) or \(x_{2}=y_{2}\) rather than just when \((x_{1},x_{2})=(y_{1},y_{2}))\); this is referred to as multiparameter SIO theory), and are multilinear rather than just linear. All these extensions require substantial technical innovation. This is particularly true for the extension from one parameter theory to multiparameter theory, because of the fact that dyadic cubes have to be replaced by much more general rectangles. Such a challenge has, so far, prevented the development of a multiparameter sparse domination theory (an alternative to representation theorems that has been extremely successful in recent years). Nevertheless, the authors show here that the proof of the representation theorem given in [\textit{A. G. De La Herrán} and \textit{T. Hytönen}, Mich. Math. J. 67, No. 4, 757--786 (2018; Zbl 1447.42016)] can be adapted to the multilinear and multiparameter setting. This proof is based on subtle groupings of multiple dyadic shifts into single objects that can be effectively estimated. The paper under review shows that this can still be done even when cubes are replaced by rectangles, and even when one needs to consider mixed operators (shifts in some components, paraproducts in others). As in [loc. cit.], the key to allowing rough kernels is to modify shifts and paraproducts, replacing projections on (tensor products of) Haar basis elements by appropriate differences of such projections. This is quite subtle. Modified shifts can be written as sums of standard shifts (Lemma 3.11), but estimating these standard shifts would require extra regularity on the kernel, so the decomposition into differences of projections is needed. The paper under review also includes various corollaries on (weighted) \(L^p\) estimates that follow naturally and fairly easily from the representation theorem. It also includes, in Section 4, commutator results that have some interesting features. For instance, the authors can decrease the regularity assumptions on the kernel when the operator has some cancelation properties (being paraproduct free). This is based on an approach devised in [\textit{K. Li} et al., Int. Math. Res. Not. 2021, No. 11, 8153--8187 (2021; Zbl 1478.42013)].
scientific article; zbMATH DE number 7446098
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