Electronic Research Announcements in Mathematical Sciences (ERA-MS)

Sharp weighted estimates for approximating dyadic operators

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

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David Cruz-Uribe, SFO - Dept. of Mathematics, Trinity College, Hartford, CT 06106-3100, United States (email)
José María Martell - Instituto de Ciencias Matemáticas CSIC-UAM-UC3M-UCM, Consejo Superior de Investigaciones Científicas, C/ Serrano 121, E-28006 Madrid, Spain (email)
Carlos Pérez - Departamento de Análisis Matemático, Facultad de Matemáticas, Universidad de Sevilla, 41080 Sevilla, Spain (email)

Abstract: 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].

Keywords:  $A_p$ weights, Haar shift operators singular integral operators, Hilbert transform, Riesz transforms, Beurling-Ahlfors operator, dyadic square function, vector-valued maximal operator.
Mathematics Subject Classification:  42B20, 42B25.

Received: January 2010;      Available Online: April 2010.