AI 中文总结
研究外尔算子的定量傅里叶限制估计中傅里叶支撑依赖性和下界。通过基于埃尔米特 - 拉盖尔对应的径向迹类估计改进上界，证明\(R\)依赖性不可避免，给出紧支撑例子的多项式下界及相关渐近最优结论。
AI 中文摘要
外尔演算将相空间\(\mathbb{R}^{2d}\)上的函数\(a\)与作用于\(L^2(\mathbb{R}^d)\)的相应外尔算子\(L_a\)相关联。在\(p = 2\)时，这种对应关系由精确的希尔伯特 - 施密特恒等式控制。对于\(p\neq2\)，已知对于帕利 - 维纳型符号的双边\(L^p\) - 沙滕估计，其常数取决于傅里叶支撑尺度。我们研究这种定量依赖性，改进已知的上界，并表明在\(p\)的大范围中不存在与支撑无关的全局比较。设\(F_{\sigma}\)表示辛傅里叶变换，\(u\in E'(\mathbb{R}^{2d})\)满足\(\text{supp}u\subset\overline{B(z_0,R)}\)，其中\(R\geq1\)。对于每个\(1\leq p\leq\infty\)和\(\varepsilon>0\)，我们证明\(\|L_{F_{\sigma} u}\|_{S_p}\lesssim_{d,p,\varepsilon}R^{(2d + 1+\varepsilon)|1 - 2/p|}\,\|F_{\sigma} u\|_{L^p(\mathbb{R}^{2d})}\)以及具有相同\(R\)幂次的反向估计。这改进了吕夫和萨穆埃尔森得到的指数依赖性\(\mathrm{e}^{cR^2}\)以及米勒的多项式依赖性\(R^{(5d + 2)|1 - 2/p|}\)。主要成分是基于埃尔米特 - 拉盖尔对应\(\rho(\varphi_k)=P_k\)的径向迹类估计，它将相关的外尔算子简化为有限秩埃尔米特投影。我们还表明对\(R\)的依赖性是不可避免的。通过截断拉盖尔函数得到的紧支撑例子给出了最佳比较常数的多项式下界。这些例子改进了米勒的算子范数例子，并为更大范围的沙滕指数给出了非平凡下界，当\(d\to\infty\)时，除了希尔伯特 - 施密特点\(p = 2\)外，可以覆盖整个范围\(1\leq p\leq\infty\)。此外，对于每个固定的\(p>2\)，当\(d\to\infty\)时，反向比较估计中的指数是渐近最优的。
英文摘要
The Weyl calculus associates a function $a$ on phase space $\mathbb{R}^{2d}$ with the corresponding Weyl operator $L_a$ acting on $L^2(\mathbb{R}^d)$. At $p=2$, this correspondence is governed by an exact Hilbert--Schmidt identity. For $p\neq2$, two-sided $L^p$--Schatten estimates are known for Paley--Wiener type symbols, with constants depending on the Fourier-support scale. We study this quantitative dependence, improve the known upper bounds, and show that in large ranges of $p$ no support-independent global comparison can hold. Let $F_σ$ denote the symplectic Fourier transform, and let $u\in E'(\mathbb{R}^{2d})$ satisfy $\operatorname{supp}u\subset\overline{B(z_0,R)}$, where $R\geq1$. Then, for every $1\leq p\leq\infty$ and $\varepsilon>0$, we prove \[ \|L_{F_σ u}\|_{S_p}\lesssim_{d,p,\varepsilon}R^{(2d+1+\varepsilon)|1-2/p|}\, \|F_σ u\|_{L^p(\mathbb{R}^{2d})}, \] together with the reverse estimate with the same power of $R$. This sharpens the exponential dependence $\mathrm{e}^{cR^2}$ obtained by Luef and Samuelsen and Müller's polynomial dependence $R^{(5d+2)|1-2/p|}$. The main ingredient is a radial trace-class estimate based on the Hermite--Laguerre correspondence $ρ(φ_k)=P_k$, which reduces the relevant Weyl operators to finite-rank Hermite projections. We also show that dependence on $R$ is unavoidable. Compactly supported examples obtained by truncating Laguerre functions yield polynomial lower bounds for the best comparison constants. These examples refine Müller's operator-norm example and give nontrivial lower bounds for a larger range of Schatten exponents, which can cover the full range $1\le p\le\infty$ except for the Hilbert--Schmidt point $p=2$ as $d\to\infty$. Moreover, for every fixed $p>2$, the exponent in the reverse comparison estimate is asymptotically optimal as $d\to\infty$.
