Vol. 6, No. 4, 2013

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Sharp weighted bounds involving $A_{\infty}$

Tuomas Hytönen and Carlos Pérez

Vol. 6 (2013), No. 4, 777–818

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]A := sup Q 1 w(Q)QM(wχQ).

In particular, we show the following improvement of the first author’s A2 theorem for Calderón–Zygmund operators T:

T(L2(w)) cT[w]A212([w] A + [w1] A)12.

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

T(Lp(w)) cpp[w] A11p([w] A)1p ,1 < p < , TfL1,(w) c[w]A1 log(e + [w]A)f L1(w).

An estimate dual to the last one is also found, as well as new bounds for commutators of singular integrals.

weighted norm inequalities, $A_p$ weights, sharp estimates, maximal function, Calderón–Zygmund operators
Mathematical Subject Classification 2010
Primary: 42B25
Secondary: 42B20, 42B35
Received: 29 July 2011
Revised: 18 November 2011
Accepted: 19 November 2011
Published: 21 August 2013
Tuomas Hytönen
Department of Mathematics and Statistics
University of Helsinki
P.O. Box 68
FI-00014 Helsinki
Carlos Pérez
Departamento de Análisis Matemático
Facultad de Matemáticas
Universidad De Sevilla
41080 Sevilla