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 $T$.Comment: We improve different parts of the first version, in particular we show the sharpness of our theorem for the vector-valued maximal functio

Weighted square function inequalities

“…See also the paper [20] by Petermichl and Pott, and [26] by Wittwer. This result was significantly extended by Cruz-Uribe, Martell, and Pérez [5] to the case of general A p weights. Here is the precise statement.…”

A $B_p$ condition for the strong maximal function

“…The first case was considered by Pérez in [26] for fractional integral operators where a fractional version of (1.8) was used to obtain a two weight L p estimate. The same problem for the Hilbert transform was proved in [5] and by different methods in [7] for any Calderón-Zygmund operator with C 1 kernel. Very recently the solution was extended in [8] to the Lipschitz case, proved full in generality in [18] and further improved in [15] with a better control on the bounds.…”

Weighted Inequalities and Dyadic Harmonic Analysis

“…It is sharp for the dyadic square function and 1 < p ≤ 2, see [21], but not for p > 2, see [48]. The optimal power for the square function is max{ 1 2 , 1 p−1 }, see [18], which corresponds to sharp extrapolation starting at r = 3 with square root power instead of starting at r = 2 with linear power, see also [50]. We conclude that sharp extrapolation is not always sharp.…”

“…Hytönen was able to prove a representation theorem valid for all Calderón-Zygmund singular integral operators (not only convolution) in terms of dyadic Haar shift operators of arbitrary complexity, paraproducts, and adjoints of the paraproducts. Different groups of researchers had already shown that the A 2 conjecture was true for all these Haar shift operators [45,17,18], using techniques other than Bellman function which had dominated the scene until then. However the dependence of the operator bound on the complexity was exponential, and prevented one from deducing the A 2 conjecture for general Calderón-Zygmund singular operators.…”