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A sparse quadratic \(T(1)\) theorem. (English) Zbl 1452.42008

Summary: We show that any Littlewood-Paley square function \(S\) satisfying a minimal Carleson condition is dominated by a sparse form, \[ \langle(S\, f)^2, g\rangle\leq C\sum_{I\in\mathcal{S}} \langle |f|\rangle^2_I \langle|g|_I|I|.\] This implies strong weighted \(L^p\) estimates for all \(A_p\) weights with sharp dependence on the \(A_p\) characteristic. In particular, the Carleson condition and the sparse domination are equivalent. The proof uses random dyadic grids, decomposition in the Haar basis, and a stopping time argument.

MSC:

42B20 Singular and oscillatory integrals (Calderón-Zygmund, etc.)
42B25 Maximal functions, Littlewood-Paley theory
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