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

Abstract

We show that two-weight L 2 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 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 p=2 (and for Young functions satisfying an appropriate integrability condition). We rely on the domination of L log ⁡ L 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).