  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.
