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arXiv · 1905.00999

Weighted Estimates of Singular Integrals and Commutators in the Zygmund Dilation Setting

Xuan Thinh Duong·Ji Li·Yumeng Ou·Jill Pipher·Brett D. Wick
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Abstract

The main purpose of this paper is to establish weighted estimates for singular integrals associated with Zygmund dilations via a discrete Littlewood--Paley theory, and then apply it to obtain the upper bound of the norm of commutators of such singular integrals with a function in the little bmo space associated with Zygmund dilations. Examples of such singular integrals associated with Zygmund dilations include a class of singular integrals studied by Ricci--Stein and Fefferman--Pipher, as well as a singular integral along a particular surface studied by Nagel--Wainger. We show that the lower bound of the norm of this commutator is not true for any singular integral in the class considered in Ricci--Stein and Fefferman--Pipher, but does in fact hold for the specific singular integral studied in Nagel--Wainger. In particular this implies that the family of singular integrals studied in these papers is not sufficiently general to contain the operator of Nagel--Wainger, which we show is of significance in this theory.

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Xuan Thinh Duong, Ji Li, Yumeng Ou, Jill Pipher, Brett D. Wick. 2019-05-03. Weighted Estimates of Singular Integrals and Commutators in the Zygmund Dilation Setting. https://arxiv.org/abs/1905.00999

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