Sharp weighted estimates for approximating dyadic operators

By D. Cruz-Uribe, J. M. Martell, C. P'erez. 2010

65 citations recorded by the source corpus.

We give a new proof of the sharp weighted $L^2$ inequality ||T||_{L^2(w)} \leq c [w]_{A_2} where $T$ is the Hilbert transform, a Riesz transform, the Beurling-Ahlfors operator or any operator that can be approximated by Haar shift operators. Our proof avoids the Bellman function technique and two weight norm inequalities. We use instead a recent result due to A. Lerner to estimate the oscillation of dyadic operators.

Subfields: Applied Mathematics

View Paper: https://www.semanticscholar.org/paper/ccd506a1151b06e8832a84d5e27ef07893d53465

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In Applied Mathematics

Citations in the corpus

A Simple Proof of the A2 Conjecture (2013, background): “[5, 6], 2010); • the L2(w) bound for general T by ‖w‖A2 log(1+ ‖w‖A2) (Pérez et al.”
A simple proof of the $A_2$ conjecture (2012, background): “This was first observed in [5, 6] (see also [17]).”
A pointwise estimate for positive dyadic shifts and some applications (2014, background): “The weighted result for the positive dyadic shifts that this contribution reduced the problem to had been already shown before in [10], see also [3] and [4].”
The A_2 theorem: Remarks and complements (2012, background): “My original proof of the A2 theorem [6] proceeded via the “top right” route corona → testing Sk → testing CZO → A2 for CZO, where the last step was borrowed from Pérez, Treil and Volberg [24].”
The sharp weighted bound for general Calderon-Zygmund operators (2010, methodology): “(The second Pérez–Treil–Volberg estimate will not be used here; it is only recorded for the sake of pointing out the connection to weak-type bounds.)”
Sharp weighted estimates of the dyadic shifts and $A_2$ conjecture (2010, background): “Another proof of the linear bound for dyadic shifts was obtained in Cruz-Uribe–Martell–Pérez [4, 5] in a very beautiful and concise approach based on a remarkable “formula” by Lerner [17].”
The A_p-A_infty inequality for general Calderon--Zygmund operators (2011, methodology): “The application of Lerner’s formula [10] to Haar shift operators is well-known, see [1], [2], [9], [6].”
SHARP BOUNDS FOR GENERAL COMMUTATORS ON WEIGHTED LEBESGUE SPACES (2010, unknown): “Recently all the above results have been recovered using different sets of techniques and obtaining linear bounds for the larger class of Haar shift operators [20], [8].”
The sharp weighted bound for general Calderón-Zygmund operators (2010, methodology): “stimate (1.4) for the good dyadic shifts in place of T. For individual shifts, this estimate has been established by Lacey–Petermichl–Reguera [13], with a simpliﬁed proof by Cruz-Uribe–Martell– Pérez [5]; however, their arguments give a dependence on certain parameters of the shift, which grows too ”
Sharp weighted estimates for classical operators (2010, unknown): “1 was announced in [6].”