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arXiv · 1001.4254

Sharp weighted estimates for classical operators

David Cruz-Uribe·Jose Maria Martell·Carlos Perez
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Abstract

We give a new proof of the sharp one weight $L^p$ inequality for any operator $T$ that can be approximated by Haar shift operators such as the Hilbert transform, any Riesz transform, the Beurling-Ahlfors operator. Our proof avoids the Bellman function technique and two weight norm inequalities. We use instead a recent result due to A. Lerner to estimate the oscillation of dyadic operators. Our method is flexible enough to prove the corresponding sharp one-weight norm inequalities for some operators of harmonic analysis: the maximal singular integrals associated to $T$, Dyadic square functions and paraproducts, and the vector-valued maximal operator of C. Fefferman-Stein. Also we can derive a very sharp two-weight bump type condition for $T$.

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David Cruz-Uribe, Jose Maria Martell, Carlos Perez. 2011-03-13. Sharp weighted estimates for classical operators. https://doi.org/10.1016/j.aim.2011.08.013

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