The main purpose of this paper is to study multi-parameter singular integral operators which commute with Zygmund dilations. We introduce a class of singular integral operators associated with Zygmund dilations and show the boundedness for these operators on L p , 1 < p < ∞, which covers those studied by and Nagel-Wainger [24].2010 Mathematics Subject Classification. 42B20, 42B30. Key words and phrases. Littlewood-Paley theory, multi-parameter singular integral operators, Zygmund dilations. R K(x 1 , x 2 , x 3 )dx 3 , as a kernel on R 2 , satisfies the regularity conditions of the classical product kernel on R 2 as studied by Fefferman and Stein in [15]. These facts, as mentioned above, can also be easily checked for singular integral operators studied
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