Sparse Domination for Bi-Parameter Operators Using Square Functions

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Let S be the dyadic bi-parameter square function Sf(x)² = ∑ᵣ ∈ D | f, hᵣ |² 1ᵣ(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 S such that | Tf, g | ∑ᵣ ∈ 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.