The sharp weighted bound for general Calderón-Zygmund operators
The sharp weighted bound for general Calderón-Zygmund operators
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DOI：
10.4007/annals.2012.175.3.9
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发表时间：
2010-07
影响因子：
4.9
通讯作者：
T. Hytönen
中科院分区：
文献类型：
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作者：
T. Hytönen
对于RN上的一般Calderon-Zygmund算子T，证明了对所有Muckenoupt权w2 A2，kTfkL2(W)C(T)sup Q A Q w Q w1 a k fkL2(W).这一最佳估计被称为A2猜想。最近Perez-Treil-Volberg的一个结果将这个问题归结为一个关于指标函数的检验条件，并在本文中得到了验证。证明包括以下元素：(I)只有一个随机系统且完全没有“坏”部分的随机并矢系统的nazarov-Treil-Volberg方法的一个变体；(Ii)作为“并矢位移”平均的一般Calderon-Zygmund算子的结果表示；以及(Iii)对这些并矢位移的Lacey-Petermichl-Rguera估计的改进，它允许在所获得的表示中对级数求和。
For a general Calderon‐Zygmund operator T on R N , it is shown that kTfkL2(w) C(T) sup Q A Q w Q w 1 a k fkL2(w) for all Muckenhoupt weights w 2 A2. This optimal estimate was known as the A2 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.