Weak and strong 𝐴_{𝑝}-𝐴_{∞} estimates for square functions and related operators

Hytönen, Tuomas; Li, Kangwei

doi:10.1090/proc/13908

Weak and strong $A_p$-$A_\infty$ estimates for square functions and related operators
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by Tuomas P. Hytönen and Kangwei Li PDF

Proc. Amer. Math. Soc. 146 (2018), 2497-2507 Request permission

Abstract:

We prove sharp weak and strong type weighted estimates for a class of dyadic operators that includes majorants of both standard singular integrals and square functions. Our main new result is the optimal bound $[w]_{A_p}^{1/p}[w]_{A_\infty }^{1/2-1/p}\lesssim [w]_{A_p}^{1/2}$ for the weak type norm of square functions on $L^p(w)$ for $p>2$; previously, such a bound was only known with a logarithmic correction. By the same approach, we also recover several related results in a streamlined manner.

References

Carme Cascante, Joaquin M. Ortega, and Igor E. Verbitsky, Nonlinear potentials and two weight trace inequalities for general dyadic and radial kernels, Indiana Univ. Math. J. 53 (2004), no. 3, 845–882. MR 2086703, DOI 10.1512/iumj.2004.53.2443
Carlos Domingo-Salazar, Michael Lacey, and Guillermo Rey, Borderline weak-type estimates for singular integrals and square functions, Bull. Lond. Math. Soc. 48 (2016), no. 1, 63–73. MR 3455749, DOI 10.1112/blms/bdv090
T. Hytönen, The sharp weighted bound for general Calderón-Zygmund operators, Ann. of Math. 175 (2012), 1473–1506.
Tuomas P. Hytönen, The $A_2$ theorem: remarks and complements, Harmonic analysis and partial differential equations, Contemp. Math., vol. 612, Amer. Math. Soc., Providence, RI, 2014, pp. 91–106. MR 3204859, DOI 10.1090/conm/612/12226
Tuomas P. Hytönen and Michael T. Lacey, The $A_p$-$A_\infty$ inequality for general Calderón-Zygmund operators, Indiana Univ. Math. J. 61 (2012), no. 6, 2041–2092. MR 3129101, DOI 10.1512/iumj.2012.61.4777
Tuomas Hytönen and Carlos Pérez, Sharp weighted bounds involving $A_\infty$, Anal. PDE 6 (2013), no. 4, 777–818. MR 3092729, DOI 10.2140/apde.2013.6.777
Michael T. Lacey, An elementary proof of the $A_2$ bound, Israel J. Math. 217 (2017), no. 1, 181–195. MR 3625108, DOI 10.1007/s11856-017-1442-x
Michael T. Lacey and Kangwei Li, On $A_p$–$A_\infty$ type estimates for square functions, Math. Z. 284 (2016), no. 3-4, 1211–1222. MR 3563275, DOI 10.1007/s00209-016-1696-8
M. Lacey, E. Sawyer, and I. Uriarte-Tuero, Two weight inequalities for discrete positive operators, available at http://arxiv.org/abs/0911.3437.
M. Lacey and J. Scurry, Weighted weak type estimates for square functions, available at http://arxiv.org/abs/1211.4219.
J. Lai, A new two weight estimates for a vector-valued positive operator, available at http://arxiv.org/abs/1503.06778.
Andrei K. Lerner, Sharp weighted norm inequalities for Littlewood-Paley operators and singular integrals, Adv. Math. 226 (2011), no. 5, 3912–3926. MR 2770437, DOI 10.1016/j.aim.2010.11.009
Andrei K. Lerner, On an estimate of Calderón-Zygmund operators by dyadic positive operators, J. Anal. Math. 121 (2013), 141–161. MR 3127380, DOI 10.1007/s11854-013-0030-1
F. Nazarov, A. Reznikov, V. Vasyunin, and A. Volberg, A Bellman function counterexample to the $A_1$ conjecture: the blow-up of the weak norm estimates of weighted singular operators, available at arXiv:1506.04710.