Improved Spectral Cluster Bounds for Orthonormal Systems

We improve Frank-Sabin’s work concerning the spectral cluster bounds for orthonormal systems at $p=\mathrm{\infty }$, on the flat torus and spaces of nonpositive sectional curvature, by shrinking the spectral band from $[{\lambda }^2,(\lambda +1{)}^2)$ to $[{\lambda }^2,(\lambda +\epsilon (\lambda ){)}^2)$, where $\epsilon (\lambda )$ is a function of $\lambda$ that goes to $0$ as $\lambda$ goes to $\mathrm{\infty }$. In achieving this, we invoke the method developed by Bourgain-Shao-Sogge-Yao.