论文标题
$ l^p $绑定了希尔伯特（Hilbert）的变换
$L^p$ bound for the Hilbert transform along variable non-flat curves
论文作者
论文摘要
我们证明了$ l^p $绑定了希尔伯特（Hilbert $α\ neqβ，\α\ neq 1，\β\ neq 1. $ 与研究情况$α=β\ neq 1 $中的相关定理相比，我们的结果更为笼统，而证明则更多。为了实现我们的目标，我们将目标函数的频率分为三种情况，并采取不同的策略来控制这些情况。此外，我们需要引入一个“短”移动最大函数$ \ mathfrak {m}^{[n]} $，以建立一些点上的估计值。
We prove the $L^p$ bound for the Hilbert transform along variable non-flat curves $(t,u(x)[t]^α+v(x)[t]^β)$, where $α$ and $β$ satisfy $α\neq β,\ α\neq 1,\ β\neq 1.$ Comparing with the associated theorem in \cite{GHLJ} investigating the case $α=β\neq 1$, our result is more general while the proof is more involved. To achieve our goal, we divide the frequency of the objective function into three cases and take different strategies to control these cases. Furthermore, we need to introduce a "short" shift maximal function $\mathfrak{M}^{[n]}$ to establish some pointwise estimate.
