Sharp A2 inequality for Haar shift operators

By Michael T. Lacey, S. Petermichl, M. Reguera. 2009

121 citations recorded by the source corpus.

AbstractAs a Corollary to the main result of the paper, we give a new proof of the inequality $$||{\rm T} f||_{L ^{2} (w)} \lesssim ||w{||_ {A_2}}||f{||_{{L ^2}(w)}}\,,$$where T is either the Hilbert transform (Amer J Math 129(5):1355–1375, 2007), a Riesz transform (Proc Amer Math Soc 136(4):1237–1249, 2008), or the Beurling operator (Duke Math J 112(2):281–305, 2002). The weight w is non-negative, and the linear growth in the A2 characteristic on the right is sharp. Prior proofs relied strongly on Haar shift operators (CR Acad Sci Paris Sér I Math 330(6):455–460, 2000) and Bellman function techniques. The new proof uses Haar shifts, and then uses an elegant ‘two weight T 1 theorem’ of Nazarov–Treil–Volberg (Math Res Lett 15(3):583–597, 2008) to immediately identify relevant Carleson measure estimates, which are in turn verified using an appropriate corona decomposition of the weight w.

Subfields: Applied Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/594c4754d8188fdfa1e8bf15d8bd64d3a2ed34be

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Citations in the corpus

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 [15], [16], [6].”
Sharp weighted bounds without testing or extrapolation (2012, background): “Most known proofs of (1) follow this paradigm, beginning with p = 2 and then extrapolating [2, 5, 10, 13, 16, 19, 20].”
Sharp weighted estimates for approximating dyadic operators (2010, methodology): “ inequality foracorresponding Haarshift operator. The norm inequalities forthese dyadic operators were then proved using Bellman function techniques. Recently, Lacey, Petermichl and Reguera-Rodriguez [14] gave a proof of the sharp A2 constant for a large family of Haar shift operators that includes ”
The A_2 theorem: Remarks and complements (2012, methodology): “[14] M. T. Lacey, S. Petermichl, and M. C. Reguera.”
The Sharp Weighted Bound for Multilinear Maximal Functions and Calderón–Zygmund Operators (2012, background): “2) Again, we refer the reader to [5,10,15,17] for the background material and further references.”
A Simple Proof of the A2 Conjecture (2013, unknown): “[15], 2010).”
Sharp weighted norm estimates beyond Calderón–Zygmund theory (2015, unknown): “This conjecture was solved by Petermichl and Volberg [2002] for the Beurling–Ahlfors transform, by Petermichl [2007; 2008] for the Hilbert and Riesz transforms (see also the alternative proof by Lacey, Petermichl and Reguera [Lacey et al. 2010]) and by Hytönen [2012] for arbitrary Calderón–Zygmund o”
Representation of singular integrals by dyadic operators, and the A_2 theorem (2011, methodology): “6 gives the following two-weight inequality, whose qualitative version was pointed out by Lacey, Petermichl and Reguera [11]: 6.”
The sharp weighted bound for general Calderon-Zygmund operators (2010, methodology): “Despite the elegance of the Cruz-Uribe–Martell– Pérez argument [5], I did not manage to modify it for the required sharpness, and the new estimates will follow instead the general outline of the original Lacey–Petermichl–Reguera proof [16].”
One and two weight norm inequalities for Riesz potentials (2012, methodology): “For the complete history of the problem we also refer the reader to [2, 18, 22, 24, 32] and the references they contain.”
The sharp weighted bound for general Calderón-Zygmund operators (2010, methodology): “Despite the elegance of the Cruz-Uribe–Martell– Pérez argument [5], I did not manage to modify it for the required sharpness, and the new estimates will follow instead the general outline of the original Lacey–Petermichl–Reguera proof [16].”
A simple proof of the $A_2$ conjecture (2012, background): “002); • Hilbert transform (S. Petermichl [22], 2007); • Riesz transform (S. Petermichl [23], 2008); • dyadic paraproduct (O. Beznosova [2], 2008); • Haar shift (M. Lacey, S. Petermichl and M. Reguera [16], 2010). After that, the following works appeared with very small intervals: • a simpliﬁed proof”