$L^p$ estimates for a singular entangled quadrilinear form
$L^p$ estimates for a singular entangled quadrilinear form
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奇异纠缠四线性形式的 $L^p$ 估计
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发表时间：
2015
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通讯作者：
Polona Durcik
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作者：
Polona Durcik
扭曲的仿积可以看作是一个二维三线性形式，它出现在Demeter和Thiele关于二维双线性Hilbert变换的工作中。$L^p$有界性的扭曲paraproduct是由于科瓦夫{c}，谁在平行建立估计并元模型的密切相关的四线性形式。通过将Kovav{c}的技巧应用到连续环境中，我们证明了后者的连续模型的一个$（L^4，L^4，L^4，L^4）$界.上述形式属于一个更大的一类具有一般调制不变性的运营商。另一个例子是三角希尔伯特变换，它控制着遍历理论中两个交换变换的相关问题，并且L^p$界仍然是一个开放的问题。
The twisted paraproduct can be viewed as a two-dimensional trilinear form which appeared in the work by Demeter and Thiele on the two-dimensional bilinear Hilbert transform. $L^p$ boundedness of the twisted paraproduct is due to Kovav{c}, who in parallel established estimates for the dyadic model of a closely related quadrilinear form. We prove an $(L^4,L^4,L^4,L^4)$ bound for the continuous model of the latter by adapting the technique of Kovav{c} to the continuous setting. The mentioned forms belong to a larger class of operators with general modulation invariance. Another instance of such is the triangular Hilbert transform, which controls issues related to two commuting transformations in ergodic theory, and for which $L^p$ bounds remain an open problem.