Improved spectral cluster bounds for orthonormal systems

We improve the work [R. L. Frank and J. Sabin, Spectral cluster bounds for orthonormal systems and oscillatory integral operators in Schatten spaces, Adv. Math. 317 (2017), 157–192] concerning the spectral cluster bounds for orthonormal systems at p = ∞, on the flat torus and spaces of nonpositive sectional curvature, by shrinking the spectral band from [λ², (λ + 1)²) to [λ², (λ + ϵ(λ))²), where ϵ(λ) is a function of λ that goes to 0 as λ goes to ∞. In achieving this, we invoke the method developed in [J. Bourgain, P. Shao, C. D. Sogge and X. Yao, On L^p-resolvent estimates and the density of eigenvalues for compact Riemannian manifolds, Comm. Math. Phys. 333 (2015), no. 3, 1483–1527].