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.

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Cited by in corpus (9)

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• Multi-parameter estimates via operator-valued shifts
• Improved surrogate bi-parameter maximum principle
• Modern singular integral theory with mild kernel regularity
• Quantitative weighted estimates for rough singular integrals on homogeneous groups
• Differences between the potential theories on a tree and on a bi-tree
• Bloom type upper bounds in the product BMO setting
• Bloom type inequality for bi-parameter singular integrals: efficient proof and iterated commutators