Spectral projector 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 the annulus

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

The spectral projector for the square root of the Euclidean Laplacian is

$$P_{\lambda,\delta}=\mathbf{1}_{(\lambda-\delta,\lambda+\delta)}(\sqrt{-\Delta}).$$

Spectral projector conjecture. If $p\geq2$ and $\kappa>0$, then, whenever $\delta>\lambda^{-1+\kappa}$,

$$\|P_{\lambda,\delta}\|_{L^2\to L^p}\lesssim_{\kappa,p}\lambda^{\frac12-\frac2p}\delta^{\frac12}+(\lambda\delta)^{\frac14-\frac{1}{2p}}.$$

Equivalently, the bound is governed by $(\lambda\delta)^{\frac14-\frac{1}{2p}}$ for $p\leq6$, and for $p\geq6$ by that term when $\delta\leq\lambda^{-1+\frac{8}{p+2}}$ and by $\lambda^{\frac12-\frac2p}\delta^{\frac12}$ when $\delta\geq\lambda^{-1+\frac{8}{p+2}}$. An $\epsilon$-loss version allows an additional factor $\lambda^\epsilon$.