Opened Access Sharp weighted bounds involving A∞

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Author: Hytönen, Tuomas
Pérez Moreno, Carlos
Department: Universidad de Sevilla. Departamento de Análisis Matemático
Date: 2013
Published in: Analysis and PDE, 6 (4), 777-818.
Document type: Article
Abstract: We improve on several weighted inequalities of recent interest by replacing a part of the Ap bounds by weaker A∞ estimates involving Wilson’s A∞ constant [w] 0 A∞ := sup Q 1 / w(Q) Z Q M(wχQ). In particular, we show the following improvement of the first author’s A2 theorem for Calderón-Zygmund operators T : kT kB(L2(w)) ≤ cT [w] 1/2 A2 [w] 0 A∞ + [w −1] 0 A∞ 1/2. Corresponding Ap type results are obtained from a new extrapolation theorem with appropriate mixed Ap A∞ bounds. This uses new two-weight estimates for the maximal function, which improve on Buckley’s classical bound.We also derive mixed A1-A∞ type results of Lerner, Ombrosi and Pérez (2009) of the form kT kB(L p(w)) ≤ cpp0 [w] 1/p A1 ([w] 0 A∞ ) 1/p 0 , 1 < p < ∞, kT f kL 1,∞(w) ≤ c[w]A1 log(e + [w] 0 A∞ )k f kL 1(w). An estimate dual to the last one is also found, as well as new bounds for commutators of singular integrals
Cite: Hytönen, T. y Pérez Moreno, C. (2013). Sharp weighted bounds involving A∞. Analysis and PDE, 6 (4), 777-818.
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DOI: 10.2140/apde.2013.6.777

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