“Small step” remodeling and counterexamples for weighted estimates with arbitrarily “smooth” weights
“Small step” remodeling and counterexamples for weighted estimates with arbitrarily “smooth” weights
复制标题
具有任意“平滑”权重的加权估计的“小步骤”重构和反例
DOI：
10.1016/j.aim.2020.107450
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发表时间：
2021
影响因子：
1.7
通讯作者：
Treil, S.
中科院分区：
文献类型：
--
作者：
Kakaroumpas, S.;Treil, S.
对于Lp（w）中希尔伯特变换的范数A p权w，1< p<∞由[w] A p s估计，其中[w] A p是权w的A p特征，s= max <$（1，1/（p− 1））;作为幂权的简单例子，这些估计是尖锐的。一个自然的问题是，是否有可能改善指数s在上述估计，如果一个取代的A p特性的“肥胖”的版本，其中的平均值被取代的泊松平均值。对于幂权重（例如p= 2和泊松平均值），可以看到指数确实有所改善：但对于一般权重是否如此？本文通过构造反例，证明了对于任意“光滑”权（在加倍常数任意接近2的意义下），最优指数s保持不变，从而“增肥”的Ap特征与经典特征等价，且使得<$T <$Lp（w）<$[w] Ap s.我们使用了F. Nazarov反驳了Sarason的猜想。我们从二元模型的简单经典反例开始，然后通过使用我们所谓的“小步构造”，我们将它们转换为具有任意二元光滑权重的示例。F. Nazarov用Bellman函数方法证明了这类例子的存在性，而我们的构造给出了一种从标准并矢例子中得到这类例子的方法。然后，我们使用的修改“重塑”，介绍了J。布尔甘和F。Nazarov，从二进模型的例子到希尔伯特变换的例子。作为额外的奖励，我们提出了一个证明，L p模拟Sarason的猜想是假的所有p，1< p<∞。
For an A p weight w the norm of the Hilbert Transform in L p (w), 1< p<∞ is estimated by [w] A p s, where [w] A p is the A p characteristic of the weight w and s= max(1, 1/(p− 1)); as simple examples with power weights show, these estimates are sharp. A natural question to ask, is whether it is possible to improve the exponent s in the above estimate if one replaces the A p characteristic by its “fattened” version, where the averages are replaced by Poisson-like averages. For power weights (for example with p= 2 and Poisson averages) one can see that there is indeed an improvement in the exponent: but is it true for general weights? In this paper we show that the optimal exponent s remains the same by constructing counterexamples for arbitrarily “smooth” weights (in the sense that the doubling constant is arbitrarily close to 2), so the “fattened” A p characteristic is equivalent to the classical one, and such that‖ T‖ L p (w)∼[w] A p s. We use the ideas from the unpublished manuscript by F. Nazarov disproving Sarason's conjecture. We start from simple classical counterexamples for dyadic models, and then by using what we call “small step construction” we transform them into examples with weights that are arbitrarily dyadically smooth. F. Nazarov had used Bellman function method to prove the existence of such examples, but our construction gives a way to get such examples from the standard dyadic ones. We then use a modification of “remodeling”, introduced by J. Bourgain and developed by F. Nazarov, to get from examples for dyadic models to examples for the Hilbert transform. As an added bonus, we present a proof that the L p analog of Sarason's conjecture is false for all p, 1< p<∞.
关于非齐次加权设置中下平方函数估计的失败
影响因子：
1.4
作者：
K. Domelevo;P. Ivanisvili;S. Petermichl;S. Treil;A. Volberg
通讯作者：
A. Volberg
影响因子：
1.7
作者：
A. Lerner
通讯作者：
A. Lerner
影响因子：
1.7
作者：
F. Nazarov;S. Petermichl;S. Treil;A. Volberg
通讯作者：
A. Volberg