Spectral projector kernel bounds on narrow windows for the two-dimensional torus

Let $\mathbb{T}^2=\mathbb{R}^2/\mathbb{Z}^2$, let $\lambda>2$ and $\delta<1$, and define

$$\mathcal{A}_{\lambda,\delta}=\{x\in\mathbb{R}^2: \lambda-\delta<|x|<\lambda+\delta\}.$$

Let

$$\Phi_{\lambda,\delta}(x)=\sum_{k\in\mathcal{A}_{\lambda,\delta}\cap\mathbb{Z}^2}e^{2\pi i k\cdot x},$$

so that $P_{\lambda,\delta}f=\Phi_{\lambda,\delta}*f$. Kernel conjecture. If $p\geq2$ and $\kappa>0$, then, whenever $\delta>\lambda^{-1+\kappa}$,

$$\|\Phi_{\lambda,\delta}\|_{L^p}\lesssim_{p,\kappa}\lambda^{1-\frac2p}\delta+(\lambda\delta)^{\frac12}.$$

Equivalently, the bound is governed by $(\lambda\delta)^{1/2}$ for $2\leq p\leq4$, by that term for $p\geq4$ and $\delta<\lambda^{4/p-1}$, and by $\lambda^{1-2/p}\delta$ for $p\geq4$ and $\delta>\lambda^{4/p-1}$. The conjecture is partially equivalent to the spectral projector conjecture and is connected to additive energies of lattice points in thin annuli.