Orthonormal Strichartz estimates on torus and waveguide manifold and applications

We establish new orthonormal Strichartz estimates for the fractional Schr\"odinger equations on torus $\mathbb T$ and waveguide manifold $\mathbb R^n\times \mathbb T^m$. We generalizes the result of Nakamura [42] on torus; while this is the first result on the waveguide manifold. The main novelty in this paper is the derivation of various kernel estimates associated to the fractional Schr\"odinger equations. Our kernel estimate generalizes the classical dispersive estimate on torus due to Kenig-Ponce-Vega [35]. On the other hand, we obtain new $\ell^2$ decoupling inequality for degeneracy type surfaces to treat the case of waveguide manifold; which maybe of independent interest and complements several known results. As an application, we establish local and small data global well-posednes for the Hartree equation with infinitely many particles with non-trace class initial data.