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 = ∞ {p=\infty} , on the flat torus and spaces of nonpositive sectional curvature, by shrinking the spectral band from [ λ 2 , ( λ + 1 ) 2 ) {[\lambda^{2},(\lambda+1)^{2})} to [ λ 2 , ( λ + ϵ ⁢ ( λ ) ) 2 ) {[\lambda^{2},(\lambda+\epsilon(\lambda))^{2})} , where ϵ ⁢ ( λ ) {\epsilon(\lambda)} is a function of λ that goes to 0 as λ goes to ∞ {\infty} . In achieving this, we invoke the method developed in [J. Bourgain, P. Shao, C. D. Sogge and X. Yao, On L p L^{p} -resolvent estimates and the density of eigenvalues for compact Riemannian manifolds, Comm. Math. Phys. 333 2015, 3, 1483–1527].