Comparison of T1 conditions for multiparameter operators

Abstract

We prove that for $L^2$ bounded operators T, the classes of operators defined in the language of vector-valued Calder\'on-Zygmund theory by Journ\'e in his proof of the T1 theorem on product spaces is of the same class as the later biparameter operators defined as the Pott and Villarroya type in their proof of the T1 theorem on product spaces.