The sharp weighted bound for general Calderon-Zygmund operators

• 成果类型: Article
• 署名作者: Hytonen, Tuomas P.
• 刊物名称: ANNALS OF MATHEMATICS
• ISSN/ISSBN: 0003-486X
• DOI: 10.4007/annals.2012.175.3.9
• 发表日期: 2012
• 页码: 1473-1506
• 关键词: ahlfors-beurling operator; SPACES

摘要

For a general Calderon Zygmund operator T on RN, it is shown that parallel to T f parallel to(L2(w)) <= C(T) . sup(Q)(f(Q) w . f(Q) w(-1)) .parallel to f parallel to(L2(w)) for all Muckenhoupt weights w is an element of A(2). This optimal estimate was known as the A(2) conjecture. A recent result of Perez-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 Calderon 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.