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

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.