One-parameter and multiparameter function classes are intersections of finitely many dyadic classes

Abstract

We prove that the class of Muckenhoupt A_p weights coincides with the intersection of finitely many suitable translates of dyadic A_p, in both the one-parameter and multiparameter cases, and that the analogous results hold for the reverse H\"older class RH_p, for doubling measures, and for the space VMO of functions of vanishing mean oscillation. We extend to the multiparameter (product) space BMO of functions of bounded mean oscillation the corresponding one-parameter BMO result due to T. Mei, by means of the Carleson-measure characterization of multiparameter BMO. Our results hold in both the compact and non-compact cases. In addition, we survey several definitions of VMO and prove their equivalences, in the continuous, dyadic, one-parameter and multiparameter cases. We show that the weighted Hardy space H^1(\omega) is the sum of finitely many suitable translates of dyadic weighted H^1(\omega), and that the weighted maximal function is pointwise comparable to the sum of finitely many dyadic weighted maximal functions for suitable translates of the dyadic grid and for each doubling weight \omega.