Multiparameter weights with connections to Schauder bases

We give several characterizations multiparameter A p ( T d ) in terms of strong maximal functions, rectangular conjugate functions, and rectangular partial sums. We also define a way to enumerate Z d that corresponds to rectangles. We then use these enumerations to give another characterization of rectangular A p ( T d ) in terms of trigonometric Schauder bases. Finally, we present some applications to principle shift invariant spaces and Gabor systems.