  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.
