The Germain–Myerson spectral projection conjecture for tori

Let $\mathbb{T}^{n}$ be the rational torus, let $P_{\lambda,\delta}$ denote its spectral projection operator for spectral parameter $\lambda$ and window width $\delta$, and let $\|P_{\lambda,\delta}\|_{2\rightarrow p}$ be its operator norm. Define

$$p_c=\frac{2(n+1)}{n-1},\qquad p^*=\frac{2n}{n-2},$$

and

$$e(p)=\frac{n+1}{n-1}\left(\frac{\frac{1}{p}-\frac{1}{p_c}}{\frac{1}{p}-\frac{n-3}{2(n-1)}}\right),$$

with

$$\mu_1(p)=\frac{n-1}{2}\left(\frac{1}{2}-\frac{1}{p}\right),\qquad \mu_2(p)=\frac{n-1}{2}-\frac{n}{p}.$$

The Germain–Myerson conjecture. If $\delta>\lambda^{-1+\kappa}$ for some fixed $\kappa\in(0,1]$, then there is a constant $C=C(n,p,\kappa,L^*)$ such that

$$\|P_{\lambda,\delta}\|_{2\rightarrow p}\leq\begin{cases} C(n,p,\kappa,L^*)(\lambda\delta)^{\frac{n-1}{2}(\frac12-\frac1p)}=C(\lambda\delta)^{\mu_1(p)},&p\leq p_c,\\ C(n,p,\kappa,L^*)(\lambda\delta)^{\frac{n-1}{2}(\frac12-\frac1p)}=C(\lambda\delta)^{\mu_1(p)},&p_c\leq p\leq p^*\text{ and }\delta\leq\lambda^{e(p)},\\ C(n,p,\kappa,L^*)\lambda^{\frac{n-1}{2}-\frac{n}{p}}\delta^{1/2}=C\lambda^{\mu_2(p)}\delta^{1/2},&p^*\leq p\text{ and }\delta\geq\lambda^{e(p)}. \end{cases}$$

These bounds refine the universal spectral projection estimates for shrinking spectral windows on the torus, down to widths just above the smallest reasonable scale for rational tori. The conjecture was originally stated by Germain and Myerson and was partially proved in the cited work; the full range of estimates remains unresolved.