$L^2$ to $L^p$ bounds for spectral projectors on the Euclidean two-dimensional torus

Abstract

We consider spectral projectors associated to the Euclidean Laplacian on the two-dimensional torus, in the case where the spectral window is narrow. Bounds for their L2 to Lp operator norm are derived, extending the classical result of Sogge; a new question on the convolution kernel of the projector is introduced. The methods employed include l2 decoupling, small cap decoupling, and estimates of exponential sums.

Figures (3)

• Figure 1: Lattice points and caps inside $\tau$
• Figure 2: The vertical axis corresponds to $\alpha=-\frac{\log \delta}{\log \lambda}$, and the horizontal axis to $\frac{1}{p}$. In the dark blue region, Conjecture \ref{['conjproj']} is verified; in the light blue region, it is verified with an $\epsilon$-loss. The red line is the curve $\delta = \lambda^{-1 + \frac{8}{p+2}}$, which separates the region where the conjecture is $\lambda^{\frac{1}{2}-\frac{2}{p}} \delta^{\frac{1}{2}}$ (below) from the region where the conjecture is $(\lambda \delta)^{\frac{1}{4}-\frac{1}{2p}}$ (above).
• Figure 3: The vertical axis corresponds to $\alpha=-\frac{\log \delta}{\log \lambda}$, and the horizontal axis to $\frac{1}{p}$. In the dark blue region, Conjecture \ref{['conjkernel']} is verified without loss; in the light blue region, it is verified with an $\epsilon$-loss. The red line is the curve $\delta = \lambda^{-1 + \frac{4}{p}}$, which separates the region where the conjecture is $\lambda^{1-\frac{2}{p}} \delta$ (below) from the region where the conjecture is $(\lambda\delta)^{\frac{1}{2}}$ (above).

Theorems & Definitions (36)

• Conjecture A: GermainMyerson1
• Remark 1.1
• Remark 1.2
• Conjecture B
• Theorem 1.3
• Theorem 1.4
• Lemma 2.1
• proof
• Lemma 2.2
• proof