Ó. Blasco and S. Pott showed that the supremum of operator norms over L2$L^2$ of all bicommutators (with the same symbol) of one‐parameter Haar multipliers dominates the biparameter dyadic product BMO norm of the symbol itself. In the present work we extend this result to the Bloom setting, and to any exponent 1
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
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).
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