2010, 17: 12-19. doi: 10.3934/era.2010.17.12

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

Received  January 2010 Published  April 2010

We give a new proof of the sharp weighted $L^p$ inequality

$ |\|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].

Citation: David Cruz-Uribe, SFO, José María Martell, Carlos Pérez. Sharp weighted estimates for approximating dyadic operators. Electronic Research Announcements, 2010, 17: 12-19. doi: 10.3934/era.2010.17.12
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