发表机构
School of Mathematics and Information Science, Guangzhou University（广州大学数学与信息科学学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究针对加权 Moran 测度建立超越维数数据的傅里叶限制精确准则，揭示最优限制范围不由标准傅里叶或维数数据决定，还构造了具有相同部分属性但临界限制指数不同的测度。
AI 中文摘要
我们为一类加权 Moran 测度建立了精确的傅里叶限制准则。假设一致有界的单-level 框架界，我们证明：对每个 \\(s>2\\)，测度本身已能检测到完整的延拓估计：\\(L^2(\mu)\to L^s(\mathbb R)\\) 延拓成立当且仅当 \\(\widehat\mu\in L^s(\mathbb R)\\)，等价于单-level 傅里叶矩 \\(G_{s,n}\\) 可和，其中 \\(G_{s,n}\\) 是第 \\(n\\) 个数字律非零傅里叶系数的 \\(s\\) 阶矩。充分性由有限框架稳定性估计及其多尺度迭代得到，而必要性则来自不相交的共振频率网格。这尤其给出了临界限制指数的精确公式和端点可达性的精确准则。该准则表明，最优限制范围并非由标准傅里叶或维数数据决定。我们构造了奇异连续谱 Moran 测度，它们具有相同的尺度、数字基数、指数谱和最大非零傅里叶系数，也满足相同的全局傅里叶衰减界，且沿同一共振频率序列达到尖锐值，但它们的临界限制指数不同。更一般地，在固定最大傅里叶系数的可和率后，我们确定了可能限制指数的最优区间，并证明在每个内部临界指数处，相同的固定数据既允许临界端点的可达，也允许其不可达。这些测度对所有容许的 \\(\theta\\) 和 \\(q\\)，可同时满足 \\( \dim_{\mathrm F}^{\theta}\mu=\theta\\) 和 \\( D_q(\mu)=1\\)，且在其支撑集上处处具有点态局部维数 1。
英文摘要
We establish an exact Fourier restriction criterion for a class of weighted Moran measures. Assuming uniformly bounded one-level frame bounds, we prove that, for every \(s>2\), the measure itself already detects the full extension estimate: \(L^2(μ)\to L^s(\mathbb R)\) extension holds if and only if \(\widehatμ\in L^s(\mathbb R)\), equivalently, if and only if the one-level Fourier moments \(G_{s,n}\) are summable, where $G_{s,n}$ is the $s$th moment of the nonzero Fourier coefficients of the $n$th digit law. The sufficient direction is obtained from a finite-frame stability estimate and its multiscale iteration, while necessity follows from disjoint resonant frequency grids. This gives, in particular, an exact formula for the critical restriction exponent and a precise criterion for endpoint attainment. The criterion reveals that the optimal restriction range is not determined by standard Fourier or dimensional data. We construct singular continuous spectral Moran measures with the same scales, digit cardinalities, exponential spectrum, and maximal nonzero Fourier coefficients. They also satisfy the same global Fourier decay bound, which is sharp along the same sequence of resonant frequencies, yet their critical restriction exponents are different. More generally, after fixing the summability rate of the maximal Fourier coefficients, we determine the optimal interval of possible restriction exponents and show that, at every interior critical exponent, the same fixed data allow both attainment and failure of the critical endpoint. These measures may simultaneously have $ \dim_{\mathrm F}^θμ=θ$ and $ D_q(μ)=1,$for all admissible $θ$ and $q$, and pointwise local dimension one everywhere on their supports.
