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

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