Subhash. R. Choudhary

arXiv2026

Refined Strichartz estimates and their orthornomal counterparts for Schrödinger equations on torus

The aim of the paper is twofold. We establish refined Strichartz estimates for the Schrödinger equation on tori within the framework of partial regularity. As a result, we reveal that the solution of the free Schrödinger equation has better regularity in mixed Lebesgue spaces. This complements the well-established theory over the past few decades, where initial data comes from the Sobolev space with respect to all spatial variables. As an application, we obtain local well-posedness for non-gauge-invariant nonlinearities with partially regular initial data. On the other hand, we extend refined Strichartz estimates for infinite systems of orthonormal functions, which generalizes the classical orthonormal Strichartz estimates on the torus by Nakamura [41] . As an application, we establish well-posedness for the Hartree equation for infinitely many fermions in some Schatten spaces. In the process, we develop several harmonic analysis tools for mixed Lebesgue spaces, e.g. Fourier multiplier transference principle, vector-valued Bernstein inequality, and vector-valued Littlewood--Paley theory for densities of operators, which may be of independent interest and complement the results of [45,55].

math.AP↗

arXiv2025

Orthonormal Strichartz estimates on torus and waveguide manifold and applications

We establish new orthonormal Strichartz estimates for the fractional Schrödinger 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 Schrödinger 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.

math.AP↗