Let $S \subset \mathbb {R}^{n}$ be a smooth compact hypersurface with a strictly positive second fundamental form, $E$ be the Fourier extension operator on $S$, and $X$ be a Lebesgue measurable subset of $\mathbb {R}^{n}$. If $X$ contains a ball of each radius, then the problem of determining the range of exponents $(p,q)$ for which the estimate $\| Ef \|_{L^{q}(X)} \lesssim \| f \|_{L^{p}(S)}$ holds is equivalent to the restriction conjecture. In this paper, we study the estimate under the following assumption on the set $X$: there is a number $0 < \alpha \leq n$ such that $|X \cap B_R| \lesssim R^{\alpha }$ for all balls $B_R$ in $\mathbb {R}^{n}$ of radius $R \geq 1$. On the left-hand side of this estimate, we are integrating the function $|Ef(x)|^{q}$ against the measure $\chi _X \,{\textrm {d}}x$. Our approach consists of replacing the characteristic function $\chi _X$ of $X$ by an appropriate weight function $H$, and studying the resulting estimate in three different regimes: small values of $\alpha$, intermediate values of $\alpha$, and large values of $\alpha$. In the first regime, we establish the estimate by using already available methods. In the second regime, we prove a weighted Hölder-type inequality that holds for general non-negative Lebesgue measurable functions on $\mathbb {R}^{n}$ and combine it with the result from the first regime. In the third regime, we borrow a recent fractal Fourier restriction theorem of Du and Zhang and combine it with the result from the second regime. In the opposite direction, the results of this paper improve on the Du–Zhang theorem in the range $0 < \alpha < n/2$.

中文翻译：

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低分形维数中的傅里叶限制

让$S \subset \mathbb {R}^{n}$是具有严格正的第二基本形式的光滑紧致超曲面，$E$成为傅里叶扩展算子$新元， 和$X$是一个 Lebesgue 可测子集$\mathbb {R}^{n}$. 如果$X$包含每个半径的球，然后是确定指数范围的问题$(p,q)$估计$\| Ef \|_{L^{q}(X)} \lesssim \| f \|_{L^{p}(S)}$等价于限制猜想。在本文中，我们研究了在以下假设下对集合的估计$X$: 有一个数字$0 < \alpha \leq n$这样$|X \cap B_R| \lesssim R^{\alpha }$所有球$B_R$在$\mathbb {R}^{n}$半径$R \geq 1$. 在这个估计的左侧，我们正在整合函数$|Ef(x)|^{q}$反对措施$\chi _X \,{\textrm {d}}x$. 我们的方法包括替换特征函数$\chi _X$的$X$通过适当的权重函数$H$，并在三种不同的情况下研究得到的估计值：$\阿尔法$, 的中间值$\阿尔法$, 和大的值$\阿尔法$. 在第一种方案中，我们使用已有的方法建立估计。在第二种情况下，我们证明了一个加权 Hölder 型不等式，它适用于一般非负 Lebesgue 可测函数$\mathbb {R}^{n}$并将其与第一种方案的结果结合起来。在第三态中，我们借用了杜和张最近的分形傅里叶限制定理，并将其与第二态的结果结合起来。在相反的方向上，本文的结果在范围内改进了 Du-Zhang 定理$0 < \alpha < n/2$.