L2 to Lp 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 L 2 to L p 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 ² decoupling, small cap decoupling and estimates of exponential sums.