Sharp weighted estimates for classical operators

Details Sharp weighted estimates for classical operators Sharp weighted estimates for classical operators

2010-01-25

arxiv.org/abs/1001.4254v2

Mathematics

Classical Analysis and ODEs

We improve different parts of the first version, in particular we show the sharpness of our theorem for the vector-valued maxi

Scientific paper

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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