Dyadic product BMO in the Bloom setting

TL;DR

This paper extends the understanding of dyadic product BMO norms and bicommutators to the Bloom setting across various exponents, using new characterizations involving paraproducts and two-weight inequalities.

Contribution

It generalizes previous results to the Bloom setting and all $1<p< inity$, introducing new characterizations and extending to multiparameter spaces.

Findings

Supremum of bicommutator norms dominates dyadic product BMO norm in Bloom setting.

Established two-weight John--Nirenberg inequalities for dyadic product BMO.

Extended results to a scale of spaces between little bmo and product BMO.

Abstract

\'O. Blasco and S. Pott showed that the supremum of operator norms over $L^2$ of all bicommutators (with the same symbol) of one-parameter Haar multipliers dominates the biparameter dyadic product BMO norm of the symbol itself. In the present work we extend this result to the Bloom setting, and to any exponent $1<p<\infty$. The main tool is a new characterization in terms of paraproducts and two-weight John--Nirenberg inequalities for dyadic product BMO in the Bloom setting. We also extend our results to the whole scale of indexed spaces between little bmo and product BMO in the general multiparameter setting, with the appropriate iterated commutator in each case.