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

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