Two-weight estimates for sparse square functions and the separated bump conjecture

Kakaroumpas, Spyridon

doi:10.1090/tran/8524

Two-weight estimates for sparse square functions and the separated bump conjecture
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Trans. Amer. Math. Soc. 375 (2022), 3003-3037 Request permission

Abstract:

We show that two-weight $L^2$ bounds for sparse square functions (uniform with respect to sparseness constants, and in both directions) do not imply a two-weight $L^2$ bound for the Hilbert transform. We present an explicit counterexample, making use of the construction due to Reguera–Thiele from [Math. Res. Lett. 19 (2012)]. At the same time, we show that such two-weight bounds for sparse square functions do not imply both separated Orlicz bump conditions on the involved weights for $p=2$ (and for Young functions satisfying an appropriate integrability condition). We rely on the domination of $L\log L$ bumps by Orlicz bumps observed by Treil–Volberg in [Adv. Math. 301 (2016), pp. 499-548] (for Young functions satisfying an appropriate integrability condition).

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