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Collectanea Mathematica

, Volume 68, Issue 1, pp 129–144 | Cite as

Two weight inequalities for bilinear forms

  • Kangwei Li
Article

Abstract

Let \(1\le p_0<p,q <q_0\le \infty \). Given a pair of weights \((w,\sigma )\) and a sparse family \({\mathcal {S}}\), we study the two weight inequality for the following bi-sublinear form
$$\begin{aligned} B(f, g)= \sum _{Q\in {\mathcal {S}}}\langle |f|^{p_0}\rangle _Q^{\frac{1}{p_0}} \langle |g|^{q_0'}\rangle _Q^{\frac{1}{q_0'}}\lambda _Q\le \mathcal N\Vert f\Vert _{L^{p}(w)}\Vert g\Vert _{L^{q'}(\sigma )}. \end{aligned}$$
When \(\lambda _Q=|Q|\) and \(p=q\), Bernicot, Frey and Petermichl showed that B(fg) dominates \(\langle Tf, g\rangle \) for a large class of singular non-kernel operators. We give a characterization for the above inequality and then obtain the mixed \(A_p-A_\infty \) estimates and the corresponding entropy bounds when \(\lambda _Q=|Q|\) and \(p=q\). We also propose a new conjecture which implies both the one supremum conjecture and the separated bump conjecture.

Keywords

\(A_p-A_\infty \) estimates Two weight theorem One supremum estimate Separated bump conjecture 

Mathematics Subject Classification

42B25 

Notes

Acknowledgments

The author would like to thank Prof. Tuomas P. Hytönen for suggesting this problem and for many helpful discussions which improve the quality of this paper.

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Copyright information

© Universitat de Barcelona 2016

Authors and Affiliations

  1. 1.Department of Mathematics and Statistics, P.O.B. 68 (Gustaf Hällströmin katu 2b)University of HelsinkiHelsinkiFinland

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