Paraproducts, Bloom BMO and Sparse BMO Functions

TL;DR

This paper investigates bounds for paraproducts in the Bloom setting, introduces sparse BMO functions, and establishes norm equivalences between sparse operators and compositions of paraproducts, advancing harmonic analysis techniques.

Contribution

It introduces sparse BMO functions associated with sparse collections and demonstrates their use in expressing sparse operators as sums of paraproducts and martingale transforms, establishing norm equivalences.

Findings

Established $L^p$ bounds for paraproducts in Bloom setting.

Introduced sparse BMO functions linked to sparse collections.

Proved norm equivalence between sparse operators and compositions of paraproducts.

Abstract

We address $L^p(\mu)\rightarrow L^p(\lambda)$ bounds for paraproducts in the Bloom setting. We introduce certain "sparse BMO" functions associated with sparse collections with no infinitely increasing chains, and use these to express sparse operators as sums of paraproducts and martingale transforms -- essentially, as Haar multipliers -- as well as to obtain an equivalence of norms between sparse operators $\mathcal{A}_\mathcal{S}$ and compositions of paraproducts $\Pi^*_a\Pi_b$.