Weak and strong type estimates for maximal truncations of Calderón-Zygmund operators on A p weighted spaces

Abstract

For 1 < p < ∞, weight w ∈ A p and any L 2 -bounded Calderón-Zygmund

The sharp weighted bound for general Calderón--Zygmund operators

“…The representation of a Calderón-Zygmund operator as an average of good dyadic shifts is an identity, which has no specific connection to A 2 weights, and may be useful for proving other bounds as well. In particular, it is likely that the same proof strategy is also applicable to providing sharp weighted weak-type L p bounds for general Calderón-Zygmund operators, in a similar way as the Lacey-Petermichl-Reguera argument was extended to weak-type L p bounds for dyadic shifts [13] and smooth Calderón-Zygmund operators [12] by Lacey et al This would involve verifying the weaktype testing condition of Lacey-Sawyer-Uriarte-Tuero [17], which is very similar to the Pérez-Treil-Volberg testing condition [26] checked in this paper. The main difference is that the Lacey-Sawyer-Uriarte-Tuero condition requires the estimation of the maximal truncations of T , rather than just the operator itself; on the other hand, the conclusions of their theorem are then valid for the maximal truncations as well.…”

“…By a different method, Lerner [19] was able to estimate all standard convolution-type operators in arbitrary dimension by controlling them in terms of Wilson's intrinsic square function [32]; however, this approach only gave (1.2) for p ∈ (1, The conjecture (1.1) concerning a strong-type bound was reduced to proving the corresponding weak-type estimate (and even slightly less) by Pérez, Treil and Volberg [26]. Based on this reduction, the first confirmation of (1.1) for a general class of nonconvolution operators, but imposing heavy smoothness requirements on the kernels, was obtained by Lacey, Reguera, Sawyer, Uriarte-Tuero, Vagharshakyan and the author [12].…”

“…The Calderón-Zygmund operator T will first be decomposed in terms of appropriate simpler operators. This was also the strategy of Lacey et al [12], where the decomposition was extracted from the proofs of the T (1) theorems due to Beylkin-Coifman-Rokhlin [2], Figiel [8], and Yang [33]. However, the mentioned decomposition seems not to have been optimal for the A 2 conjecture, as summing up the weighted estimates for the simple operators required a high degree of smoothness on the kernel of T .…”

“…Despite the elegance of the Cruz-Uribe-MartellPérez argument [5], I did not manage to modify it for the required sharpness, and the new estimates will follow instead the general outline of the original Lacey-Petermichl-Reguera proof [16]. in the sharp weighted inequalities through my participation in the joint efforts [12], [13]. The financial support of the Academy of Finland through projects 130166, 133264 and 218418 is gratefully acknowledged.…”

Weighted L p estimates for the area integral associated to self-adjoint operators

“…Note that sharp weighted optimal bounds for singular integrals has been studied extensively, see for examples, [14,24,28,29,31] and the references therein. One then has the analogous statement as in Theorems 1.1, 1.2, 1.3 and 1.4 replacing s p , s h , S P , S H by g p , g h , G P , G H , respectively.…”