Abstract
As a Corollary to the main result of the paper, we give a new proof of the inequality
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 A 2 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.
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Research of M. T. Lacey and M. C. Reguera supported in part by NSF grant 0456611.
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Lacey, M.T., Petermichl, S. & Reguera, M.C. Sharp A 2 inequality for Haar shift operators. Math. Ann. 348, 127–141 (2010). https://doi.org/10.1007/s00208-009-0473-y
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DOI: https://doi.org/10.1007/s00208-009-0473-y