Sharp weighted estimates for classical operators

Abstract

We prove sharp one and two-weight norm inequalities for some of the classical operators of harmonic analysis: the Hilbert and Riesz transforms, the Beurling-Ahlfors operator, the maximal singular integrals associated to these operators, the dyadic square function and the vector-valued maximal operator. In the twoweight case we prove that these operators map L p (v) into L p (u), 1 < p < ∞, provided that the pair (u, v) satisfies the Ap bump condition sup Q ku 1/pkA,Qkv −1/pkB,Q < ∞, where A¯ ∈ Bp0 and B¯ ∈ Bp. These conditions are known to be sharp in many cases and they characterize, in the scale of these Ap bump conditions, the corresponding two-weight norm inequalities for the Hardy-Littlewood maximal operator M: i.e., M : L p (v) −→ L p (u) and M : L p 0 (u 1−p 0 ) −→ L p (v 1−p 0 ). All of these results give positive answers to conjectures we made in. In the one-weight case we prove the sharp dependence on the Ap constant by finding the best value for the exponent α(p) such that kT fkLp(w) ≤ Cn,T [w] α(p) Ap kfkLp(w) For the Hilbert transform, the Riesz transforms and the BeurlingAhlfors operator the sharp value of α(p) was found by Petermichl and Volberg; their proofs used approximations by the dyadic Haar shift operators, Bellman function techniques, and twoweight norm inequalities. Our results for dyadic square functions and vector-valued maximal operators are new. All of our proofs again depend on dyadic approximation, but avoid Bellman functions and two-weight norm inequalities. We instead use a recent result due to A. Lerner [30] to estimate the oscillation of dyadic operators. A key feature of our approach is that it will extend to any operator that can be approximated by Haar shift operators.