Sharp weighted estimates for classical operators

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 $T$.

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

References in corpus (5)

• A characterization of two weight norm inequalities for maximal singular integrals with one doubling measure
• Linear bound for the dyadic paraproduct on weighted Lebesgue space $L_2(w)$
• On $A_2$ conjecture and corona decomposition of weights
• Sharp $A_2$ Inequality for Haar Shift Operators
• Weak and Strong-type estimates for Haar Shift Operators: Sharp power on the $A_p$ characteristic

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