2019/05/03 by Xuan Thinh Duong, Duong, Xuan Thinh, Ji Li +7
Mathematics · #Advanced Harmonic Analysis Research #Advanced Mathematical Physics Problems #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Mathematical Analysis and Transform Methods
paper · pdf · doi:10.48550/arxiv.1905.00999
openalex publication_date 2019/05/03 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.