$H^1$ and dyadic $H^1$

Abstract

In this paper we give a simple proof of the fact that the average over all dyadic lattices of the dyadic $H^1$-norm of a function gives an equivalent $H^1$-norm. The proof we present works for both one-parameter and multi-parameter Hardy spaces. The results of such type are known. The first result (for one-parameter Hardy spces) belongs to Burgess Davis (1980). Also, by duality, such results are equivalent to the "BMO from dyadic BMO" statements proved by Garnett-Jones(1982} for one parameter case, and by Pipher-Ward (2008) for two-parameter case. While the paper generalizes these results to the multi-parameter setting, this is not its main goal. The purpose of the paper is to present an approach leading to a simple proof, which works in both one-parameter and multi-parameter cases. The main idea of treating square function as a Calderon--Zygmind operator is a commonplace in harmonic analysis; the main observation, on which the paper is based, is that one can treat the random dyadic square function this way. After that, all is proved by using the standard and well-known results about Calderon--Zygmind operators in the Hilbert-space-valued setting. As an added bonus, we get a simple proof of the (equivalent by duality) inclusion $\text{BMO}\subset \text{BMO}_d$, $H^1_d \subset H^1$ in the multi-parameter case. Note, that unlike the one-parameter case, the inclusions in the general situation are far from trivial.