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 $p\neq2$, 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 $F_{\sigma}$ 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 \[ \|L_{F_{\sigma} u}\|_{S_p}\lesssim_{d,p,\varepsilon}R^{(2d+1+\varepsilon)|1-2/p|}\, \|F_{\sigma} u\|_{L^p(\mathbb{R}^{2d})}, \] 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 $\rho(\varphi_k)=P_k$, 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 $d\to\infty$.