Product space singular integrals with mild kernel regularity
type: Item
instance of: scholarly article
title: Product space singular integrals with mild kernel regularity
zbMATH Open document ID: 1491.42013
author: Emil Airta
author: Henri Martikainen
author: Emil Vuorinen
published in: The Journal of Geometric Analysis
publication date: 2021-12-15T00:00:00Z
review text: \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, pa…
reviewed by: Pierre Portal
Mathematics Subject Classification ID: 42B20
zbMATH DE Number: 7446098
zbMATH Keywords: singular integrals
zbMATH Keywords: commutators
zbMATH Keywords: weighted estimates
zbMATH Keywords: kernel regularity
zbMATH Keywords: multilinear analysis
zbMATH Keywords: multi-parameter analysis
Wikidata QID: Q114221030
MaRDI profile type: Publication
full work available at URL: s12220-021-00757-3
OpenAlex ID: W4200329874
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DOI: 10.1007/S12220-021-00757-3
description: scientific article; zbMATH DE number 7446098