Multiparameter weights with connections to Schauder bases

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