The sharp weighted bound for general Calderón—Zygmund operators

For a general Calderón—Zygmund operator T on ℝ N , it is shown that $\normal{||Tf||_{L^{2}(w)} \le C(T) \cdot \underset {Q}\sup} \LARGE{(} \Huge{f}\normal{_{Q^{w}} \cdot} \Huge{f}\normal{_{Q^{w}}}\small{^\{-1}}\LARGE{)}\normal{\cdot ||f||_{L^{2}(w)}}$ for all Muckenhoupt weights w ∈ A 2 . This optim...

Full description

Saved in:
Bibliographic Details
Published in:Annals of mathematics 2012-05, Vol.175 (3), p.1473-1506
Main Author: Hytönen, Tuomas P.
Format: Article
Language:English
Subjects:
Conditional probabilities
Cubes
Dyadics
Martingales
Mathematical functions
Mathematical inequalities
Mathematical integrals
Mathematical theorems
Preprints
Citations: Items that this one cites
Items that cite this one
Online Access:Get full text
Tags: Add Tag
No Tags, Be the first to tag this record!
Description
Summary:For a general Calderón—Zygmund operator T on ℝ N , it is shown that $\normal{||Tf||_{L^{2}(w)} \le C(T) \cdot \underset {Q}\sup} \LARGE{(} \Huge{f}\normal{_{Q^{w}} \cdot} \Huge{f}\normal{_{Q^{w}}}\small{^\{-1}}\LARGE{)}\normal{\cdot ||f||_{L^{2}(w)}}$ for all Muckenhoupt weights w ∈ A 2 . This optimal estimate was known as the A 2 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 Nazarov—Treil—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.
ISSN:0003-486X
DOI:10.4007/annals.2012.175.3.9