Quantitative Fourier Restriction Estimates for Weyl Operators: Fourier-Support Dependence and Lower Bounds

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 $p\mathrm{\ne }2$, 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 $F_{\sigma }$ denote the symplectic Fourier transform, and let $u\in E^{\mathrm{\prime }}({\mathbb{R}}^{2d})$ satisfy $\mathrm{supp}u\subset \overline{B(z_0,R)}$, where $R\ge 1$. Then, for every $1\le p\le \mathrm{\infty }$ and $\epsilon >0$, we prove $$\mathrm{\parallel }{L}_{F_{\sigma }u}{\mathrm{\parallel }}_{S_p}{\lesssim }_{d,p,\epsilon }{R}^{(2d+1+\epsilon )\mathrm{\mid }1-2\mathrm{/}p\mathrm{\mid }}\mathrm{\parallel }{F}_{\sigma }u{\mathrm{\parallel }}_{L^p({\mathbb{R}}^{2d})},$$ together with the reverse estimate with the same power of $R$. This sharpens the exponential dependence ${\mathrm{e}}^{c{R}^2}$ obtained by Luef and Samuelsen and Müller’s polynomial dependence $R^{(5d+2)\mathrm{\mid }1-2\mathrm{/}p\mathrm{\mid }}$. The main ingredient is a radial trace-class estimate based on the Hermite–Laguerre correspondence $\rho ({\phi }_k)=P_k$, 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 $1\le p\le \mathrm{\infty }$ except for the Hilbert–Schmidt point $p=2$ as $d\to \mathrm{\infty }$. Moreover, for every fixed $p>2$, the exponent in the reverse comparison estimate is asymptotically optimal as $d\to \mathrm{\infty }$.