Sharp weighted estimates for approximating dyadic operators

David Cruz-Uribe, SFO; José María Martell; Carlos Pérez

doi:10.3934/era.2010.17.12

Article Contents

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

This volume Previous Article Multifractal formalism derived from thermodynamics for general dynamical systems Next Article 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
2.
Instituto de Ciencias Matemáticas CSIC-UAM-UC3M-UCM, Consejo Superior de Investigaciones Científicas, C/ Serrano 121, E-28006 Madrid
3.
Departamento de Análisis Matemático, Facultad de Matemáticas, Universidad de Sevilla, 41080 Sevilla

Received: December 31, 2009

Published: December 2009

Abstract Related Papers Cited by

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.

Citation:

Article Metrics

HTML views() PDF downloads(100) Cited by(0)

Sharp weighted estimates for approximating dyadic operators

Abstract

Access History

Article Metrics

Access History

Other Articles By Authors

Catalog

Sharp weighted estimates for approximating dyadic operators

Abstract

Access History

SHARE

Article Metrics

Access History

Other Articles By Authors

Catalog

Export File

Citation

Format

Content