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

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- by Spyridon Kakaroumpas PDF
- 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).## References

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## Additional Information

**Spyridon Kakaroumpas**- Affiliation: Department of Mathematics, Brown University, Providence, Rhode Island
- Address at time of publication: Institut für Mathematik, Julius-Maximilians-Universität Würzburg, 97074 Würzburg, Germany.
- MR Author ID: 1331246
- Received by editor(s): January 23, 2020
- Received by editor(s) in revised form: March 19, 2021
- Published electronically: February 9, 2022
- © Copyright 2022 American Mathematical Society
- Journal: Trans. Amer. Math. Soc.
**375**(2022), 3003-3037 - MSC (2020): Primary 42B20
- DOI: https://doi.org/10.1090/tran/8524
- MathSciNet review: 4402654