The sharp weighted bound for general Calderón--Zygmund operators

Abstract

For a general Calderón-Zygmund operator T on R N , it is shown thatfor all Muckenhoupt weights w ∈ A2. This optimal estimate was known as the A2 conjecture. A recent result of Pérez-Treil-Volberg reduced the problem to a testing condition on indicator functions, which is verified in this paper. The proof consists of the following elements: (i) a variant of the NazarovTreil-Volberg method of random dyadic systems with just one random system and completely without "bad" parts; (ii) a resulting representation of a general Calderón-Zygmund operator as an average of "dyadic shifts;" and (iii) improvements of the Lacey-Petermichl-Reguera estimates for these dyadic shifts, which allow summing up the series in the obtained representation.

Sharp weighted bounds for the q -variation of singular integrals

“…However, their arguments used the random dyadic systems, and were derived from the (non-positive) dyadic representation theorem of [3]. Here we give a more efficient direct argument.…”

“…The following sharp weighted inequality for Calderón-Zygmund operators, known as the A 2 theorem, was proved by the first author [3] in full generality after many intermediate results by others:…”

“…However, all of them have been based on the "dyadic representation theorem" from [3], namely, the probabilistic formula…”

On general local $Tb$ theorems

“…There are subtle tricks which depend on independence to add and remove goodness; see [Hyt09], [Hyt10b] and [Mar10]. These cannot be used here either, and this is basically because Δ Q depends not only on the cube Q and its children (as in the global T b theorems), but also, through the stopping time argument, on the whole grid D (and this stops one from using certain independence properties).…”

Weighted Bounds for Multilinear Square Functions

“…In 2012, T. Hytönen [19] proved the so-called A 2 theorem, which asserted that the sharp dependence of the L 2 (w) norm of a Calderón-Zygmund operator on the A 2 constant of the weight w was linear. More precisely,…”