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 . Contents 1. Introduction and main results 1 2. Overview of the paper 6 3. The Reguera-Thiele [18] construction 10 4. Two-weight estimates for generalized sparse operators 13 5. Investigating separated bump conditions 25 6. Appendix 33 References 35