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

TL;DR

This paper demonstrates that various function classes like Muckenhoupt A_p weights, reverse H"older classes, and BMO can be represented as intersections or sums of finitely many dyadic classes, extending known results to multiparameter and weighted settings.

Contribution

It extends the characterization of function classes as intersections of dyadic classes to multiparameter, weighted, and vanishing mean oscillation cases, providing new unified frameworks.

Findings

A_p weights are intersections of finitely many dyadic A_p classes.

Weighted Hardy space H^1(ω) is the sum of finitely many dyadic weighted H^1(ω) spaces.

Weighted maximal functions are comparable to sums of dyadic maximal functions.

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.