Sparse Domination for Bi-Parameter Operators Using Square Functions

Abstract

Let $S$ be the dyadic bi-parameter square function $$Sf(x{)}^2=\sum _{R\in \mathcal{D}}|⟨f,h_R⟩{|}^2\frac{1_R(x)}{|R|}.$$ We prove that if $T$ is a bi-parameter martingale transform and $f,g$ are suitable test functions, then there exists a sparse collection of rectangles $\mathcal{S}$ such that $$|⟨Tf,g⟩|\lesssim \sum _{R\in \mathcal{S}}|R|(Sf{)}_R(Sg{)}_R.$$ We also extend this estimate to the case where $T$ is a bi-parameter cancellative dyadic shift and when $T$ is a paraproduct-free singular integral of Journé type. Weighted estimates follow from the domination.

References in corpus (1)

• Sharp Reverse Hölder property for A_\infty weights on spaces of homogeneous type