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

Abstract

We show that two-weight $L^2$ bounds for sparse square functions, uniformly with respect to the sparseness constant of the underlying sparse family, and in both directions, do not imply a two-weight $L^2$ bound for the Hilbert transform. We present an explicit example, making use of the construction due to Reguera--Thiele from [18]. At the same time, we show that such two-weight bounds for sparse square functions do not imply both separated Orlicz bump conditions of 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 (for Young functions satisfying an appropriate integrability condition) observed by Treil--Volberg in [20].