×

A Bellman function proof of the \(L^2\) bump conjecture. (English) Zbl 1284.42037

The original (still) open problem concerning two weight estimates for singular integrals is to find necessary and sufficient condition on the weights \(u\) and \(v\) such that a Calderón-Zygmund operator \(T: L^p(u) \to L^p(v)\) is bounded, i.e., \[ \int_{\mathbb R^n} | Tf |^p v \, dx \leq C \int_{\mathbb R^n} | f |^p u \, dx. \] We can rewrite the problem as follows. Describe all weights \(u,v\) such that the operator \(M_{v^{1/p}} T M_{u^{1/p'}}\) is bounded on non-weighted \(L^p\), where \(M_a\) is the multiplication operator \(M_a f= af.\)
The \(A_p\) condition \[ \sup_{I} \Big( \frac{1}{ | I |} \int_I v \, dx \Big)^{1/p} \Big( \frac{1}{ | I |} \int_I u \, dx \Big)^{1/p'} < \infty \] is necessary for the boundedness of the operator \(M_{v^{1/p}} T M_{u^{1/p'}}\), but simple counterexamples show that this condition is not sufficient. So a natural way to obtain a sufficient condition is to replace the \(L^1\) norm of \(u\) and \(v\) with some stronger Orlicz norms. Namely, given a Young function \(\Phi\) and a cube \(I\), one can consider the Orlicz space \(L^{\Phi}(I)\) with norm given by \[ \| f \|_{L^{\Phi}(I)} := \inf \left\{ \lambda>0 : \;\int_I \Phi \left( \frac{f(x)}{ \lambda} \right) \frac{dx}{ | I |} \leq 1 \right\}. \] It has been conjectured (for \(p=2\)) that if the Young functions \(\Phi_1\) and \(\Phi_2\) satisfy some suitable conditions, then \[ \sup_{I} \| v \|_{\Psi_1 (I)} \| u \|_{\Psi_2 (I)} < \infty \] implies that \(M_{v^{1/2}} T M_{u^{1/2}}\) is bounded on \(L^2\).
A. K. Lerner [J. Anal. Math. 121, 141–161 (2013; Zbl 1285.42015)] solved this problem. In this paper the authors give another proof by using a Bellman function.
Define the normalized distribution function \(N\) by \[ N_I^{w}(t) = \frac{1}{ | I |} \left| \{ x \in I : w(x) >t\} \right|. \] Let \(\Psi : (0,1] \to \mathbb R_{+}\) be a decreasing function such that the function \(s \to s \Psi (s)\) is increasing. Let \(\Phi\) be a Young function and \(\Psi (s) \leq C \Phi'(t) \) where \(s = 1/\Psi(t) \Psi'(t)\) for all sufficiently large \(t\). Then \[ n_{\Psi}(N_I^{w}):= \int_0^{\infty} N_I^{w}(t) \Psi (N_I^{w}(t))dt \leq C \| w \|_{L^{\Phi}(I)}. \] They prove the following: If \[ \sup_{I} n_{\Psi_1}(N_I^{v}) n_{\Psi_2}(N_I^{u}) < \infty, \] then \(M_{v^{1/2}} T M_{u^{1/2}}\) is bounded on \(L^2\). For the proof they use the ideas of dyadic shifts and paraproducts by T. Hytönen, C. Pérez, S. Treil and A. Volberg [“Sharp weighted estimates for dyadic shifts and the \(A_2\) conjecture”, J. Reine Angew. Math. (2014)].

MSC:

42B20 Singular and oscillatory integrals (Calderón-Zygmund, etc.)
42B25 Maximal functions, Littlewood-Paley theory

Citations:

Zbl 1285.42015
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] D. Cruz-Uribe, J. M. Martell, and C. Pérez, Sharp two-weight inequalities for singular integrals, with applications to the Hilbert transform and the Sarason conjecture, Adv. Math. 216 (2007), 647–676. · Zbl 1129.42007 · doi:10.1016/j.aim.2007.05.022
[2] D. Cruz-Uribe, J. M. Martell, and C. Pérez, Weights, extrapolation and the theory of Rubio de Francia, Birkhäuser/Springer Basel AG, Basel, 2011. · Zbl 1234.46003
[3] D. Cruz-Uribe, J. M. Martell, and C. Pérez, Sharp weighted estimates for classical operators, Adv. Math. 229 (2012), 408–441. · Zbl 1236.42010 · doi:10.1016/j.aim.2011.08.013
[4] D. Cruz-Uribe and C. Pérez, Sharp two-weight, weak-type norm inequalities for singular integral operators, Math. Res. Lett. 6(1989), 417–427. · Zbl 0961.42013 · doi:10.4310/MRL.1999.v6.n4.a4
[5] D. Cruz-Uribe and C. Pérez, Two-weight, weak-type norm inequalities for fractional integrals, Calderón-Zygmund operators and commutators, Indiana Univ. Math. J. 49 (2000), 697–721. · Zbl 1033.42009 · doi:10.1512/iumj.2000.49.1795
[6] D. Cruz-Uribe and C. Pérez, On the two-weight problem for singular integral operators, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 1 (2002), 821–849. · Zbl 1072.42010
[7] D. Cruz-Uribe, A. Reznikov, and A. Volberg, Logarithmic bump conditions and the two-weight boundedness of Calderón-Zygmund operators, arXiv:1112. 0676v4, [math. AP]. · Zbl 1290.42033
[8] C. Fefferman, The uncertainty principle, Bull. Amer. Math. Soc. 9 (1983), 129–206. · Zbl 0526.35080 · doi:10.1090/S0273-0979-1983-15154-6
[9] T. Hytönen,, The sharp weighted bound for general Calderón-Zygmund operators, Ann. of Math. (2), 175 (2012), 1473–1506. · Zbl 1250.42036 · doi:10.4007/annals.2012.175.3.9
[10] T. Hytönen, C. Pérez, S. Treil, and A. Volberg,, Sharp weighted estimates for dyadic shifts and the A 2 conjecture, J. Reine Angew. Math., to appear; arXiv:11010. 0755v2, [math. CA]. · Zbl 1311.42037
[11] P. Koosis, Moyennes quadratiques pondérées de fonctions périodiques et de leurs conjuguées harmoniques, C. R. Acad. Sci. Paris Sér. A-B 291 (1980), A255–A257. · Zbl 0454.42009
[12] A. Lerner, A pointwise estimate for local sharp maximal function with applications to singular integrals, Bull. London Math. Soc. 42 (2010), 843–856. · Zbl 1203.42023 · doi:10.1112/blms/bdq042
[13] A. Lerner, On an estimate of Calderón-Zygmund operators by dyadic positive operators, J. Anal. Math. 121 (2013), 141–161. · Zbl 1285.42015 · doi:10.1007/s11854-013-0030-1
[14] C. Liaw and S. Treil, Regularizations of general singular integral operators, Rev. Mat. Iberoam. 29 (2013), 53–74. · Zbl 1272.42011 · doi:10.4171/RMI/712
[15] F. Nazarov, A. Reznikov, S. Treil, and A. Volberg, The sharp bump condition for the two-weight problem for classical singular integral operator: the Bellman function approach, preprint, October, 2011.
[16] F. Nazarov, A. Reznikov, S. Treil, and A. Volberg, A solution of the bump conjecture for all Calderón-Zygmund operators: the Bellman function approach, sashavolberg. wordpress. com, Feb. 11, 2012.
[17] F. Nazarov, A. Reznikov, and A. Volberg, The proof of A 2 conjecture in a geometrically doubling metric space, Indiana Univ. Math. J. to appear. arXiv:1106. 1342v2 [math. CA]. · Zbl 1293.42017
[18] F. Nazarov, S. Treil, and A. Volberg, Bellman function and two-weight inequality for martingale transform, J. Amer. Math. Soc. 12 (1999), 909–928. · Zbl 0951.42007 · doi:10.1090/S0894-0347-99-00310-0
[19] C. J. Neugebauer, Inserting A p-weights, Proc. Amer. Math. Soc. 87 (1983), 644–648. · Zbl 0521.42019
[20] C. Pérez, Weighted norm inequalities for singular integral operators, J. LondonMath. Soc.(2) 49 (1994), 296–308. · Zbl 0797.42010 · doi:10.1112/jlms/49.2.296
[21] C. Pérez, On sufficient conditions for the boundedness of the Hardy-Littlewood maximal operator between weighted Lp-spaces with different weights, Proc. LondonMath. Soc.(3) 71 (1995), 135–157. · Zbl 0829.42019 · doi:10.1112/plms/s3-71.1.135
[22] C. Pérez and R. Wheeden, Uncertainty principle estimates for vector fields, J. Funct. Anal. 181 (2001), 146–188. · Zbl 0982.42010 · doi:10.1006/jfan.2000.3711
[23] J. L. Rubio de Francia, Boundedness of maximal functions and singular integrals in weighted Lp spaces, Proc. Amer. Math. Soc. 83 (1981), 673–679. · Zbl 0477.42011 · doi:10.2307/2044232
[24] S. Treil, Sharp A 2 estimates of Haar shifts via Bellman function, arXiv:1105. 2252v1 [math. CA].
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.