Improved spectral projection estimates

We obtain new improved spectral projection estimates on manifolds of non-positive curvature, including sharp ones for relatively large spectral windows for general tori. Our results are stronger than those in an earlier work of the first and third authors (2019), and the arguments have been greatly simplified. We more directly make use of pointwise estimates that are implicit in the work of Bérard (1977) and avoid the use of weak-type spaces that were used in the previous works by the first and third authors (2017, 2019). We also simplify and strengthen the bilinear arguments by exploiting the use of microlocal Kakeya–Nikodym L^{2}\to L^{q_c} -estimates and avoiding the use of L^{2}\to L^{2} ones as in earlier results. This allows us to prove new results for manifolds of negative curvature and some new sharp estimates for tori. We also have new and improved techniques in two dimensions for general manifolds of non-positive curvature.