Dyadic structure theorems for multiparameter function spaces

Abstract

We prove that the multiparameter (product) space BMO of functions of bounded mean oscillation can be written as the intersection of finitely many dyadic product BMO spaces, with equivalent norms, generalizing the one-parameter result of T. Mei. We establish the analogous dyadic structure theorems for the space VMO of functions of vanishing mean oscillation, for $A_p$ weights, for reverse-Hölder weights and for doubling weights. We survey several definitions of VMO and prove their equivalences, in the continuous, dyadic, one-parameter and product cases. In particular, we introduce the space of dyadic product VMO functions. We show that the weighted product Hardy space $H_{\omega }^1$ is the sum of finitely many translates of dyadic weighted $H_{\omega }^1$ , for each $A_{\infty }$ weight $\omega$, and that the weighted strong maximal function is pointwise comparable to the sum of finitely many dyadic weighted strong maximal functions, for each doubling weight $\omega$. Our results hold in both the compact and non-compact cases.