论文标题
谐波分析中的离散类似物：Stein-Wainger的定理
Discrete Analogues in Harmonic Analysis: A Theorem of Stein-Wainger
论文作者
论文摘要
对于$ d \ geq 2，\ d \ geq 1 $，令$ \ m rathscr {p} _ {d，d} $表示所有度$ d $ polyenmials的集合，$ d $ d $ dopyenmials，带有实际系数，没有线性项。我们证明，对于任何calderón-zygmund内核，$ k $，最大调制和最大截断的离散单数积分操作员，\ begin {align*} \ sup_ {p \ in \ mathscr {p} \ sum_ {0 <| m | \ leq n} f（x-m）k（m）e^{2πip（m）} \ big |，\ end {align*}在$ \ ell^p（\ mathbb {z}^d）$上限制在$ \ ell^p（\ ell^p（\ ell^p）上。我们的证明是根据多项式轨道的等分分配理论引入了停止时间，以将分析与其连续类似物相关联，由Stein-Wainger引入和研究：\ begin {align*} \ sup_ {p \ in \ mathscr in \ mathscr {p} _ {p} _ {d，d，d，d，d，d，d，d，d，d，d，d，d，d，big | \ int _ {\ mathbb {r}^d} f（x-t）k（t）e^{2πip（t）} \ dt \ big |。 \ end {align*}
For $d \geq 2, \ D \geq 1$, let $\mathscr{P}_{d,D}$ denote the set of all degree $d$ polynomials in $D$ dimensions with real coefficients without linear terms. We prove that for any Calderón-Zygmund kernel, $K$, the maximally modulated and maximally truncated discrete singular integral operator, \begin{align*} \sup_{P \in \mathscr{P}_{d,D}, \ N} \Big| \sum_{0 < |m| \leq N} f(x-m) K(m) e^{2πi P(m)} \Big|, \end{align*} is bounded on $\ell^p(\mathbb{Z}^D)$, for each $1 < p < \infty$. Our proof introduces a stopping time based off of equidistribution theory of polynomial orbits to relate the analysis to its continuous analogue, introduced and studied by Stein-Wainger: \begin{align*} \sup_{P \in \mathscr{P}_{d,D}} \Big| \int_{\mathbb{R}^D} f(x-t) K(t) e^{2πi P(t)} \ dt \Big|. \end{align*}
