论文标题
多分辨率分析和Zygmund扩张
Multiresolution analysis and Zygmund dilations
论文作者
论文摘要
Zygmund的扩张是标准产品理论和单参数设置之间的一组扩张 - 在$ \ Mathbb {r}^3 = \ Mathbb {r} \ times \ times \ times \ mathbb {r} \ times \ times \ times \ mathbb {r} $中，它们是限制$（x_1，x_1，x_2，x_2，x_2，x_2，x_2，x_2，x_2，x_2，x_2，x3）。 x_2，δ_1Δ_2x_3）$。在现代产物奇异的积分理论中，二元性多解析分析和相关的二元核能方法非常有影响力。但是，在Zygmund扩张设置或其他修改的产品空间设置中尚未了解多分辨率分析。在本文中，我们开发了Zygmund类型的这种缺失的二元多解析分析，并通过在加权空间上界定其有用性是合理的，在Zygmund膨胀下是不变的一般单数积分。我们提供了Zygmund $ a_p $ striges和Zygmund内核的新颖示例，以展示我们内核假设的最佳性以进行加权估计。
Zygmund dilations are a group of dilations lying in between the standard product theory and the one-parameter setting - in $\mathbb{R}^3 = \mathbb{R} \times \mathbb{R} \times \mathbb{R}$ they are the dilations $(x_1, x_2, x_3) \mapsto (δ_1 x_1, δ_2 x_2, δ_1 δ_2 x_3)$. The dyadic multiresolution analysis and the related dyadic-probabilistic methods have been very impactful in the modern product singular integral theory. However, the multiresolution analysis has not been understood in the Zygmund dilation setting or in other modified product space settings. In this paper we develop this missing dyadic multiresolution analysis of Zygmund type, and justify its usefulness by bounding, on weighted spaces, a general class of singular integrals that are invariant under Zygmund dilations. We provide novel examples of Zygmund $A_p$ weights and Zygmund kernels showcasing the optimality of our kernel assumptions for weighted estimates.
