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Theory of $(a,b)$-continued fraction transformations and applications
Sharp weighted estimates for approximating dyadic operators
| 1. | Dept. of Mathematics, Trinity College, Hartford, CT 06106-3100, United States |
| 2. | Instituto de Ciencias Matemáticas CSIC-UAM-UC3M-UCM, Consejo Superior de Investigaciones Científicas, C/ Serrano 121, E-28006 Madrid, Spain |
| 3. | Departamento de Análisis Matemático, Facultad de Matemáticas, Universidad de Sevilla, 41080 Sevilla, Spain |
$ |\|T\||_{L^p(w)} \leq C_{n,T}[w]_{A_p}^{\max(1,\frac{1}{p-1})}, $
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 [15] to estimate the
oscillation of dyadic operators.
The method we use is flexible enough to obtain the sharp one-weight
result for other important operators as well as a very sharp
two-weight bump type result for $T$ as can be found in
[5].
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