# The Hartree-Fock equations in modulation spaces

## TL;DR

This paper develops local and global well-posedness theories for nonlinear Hartree-Fock equations within modulation spaces, including cases with harmonic potentials, and proves boundedness of multilinear operators on these spaces.

## Contribution

It introduces well-posedness results for Hartree-Fock equations in modulation spaces and extends these results to equations with harmonic potentials, along with proving multilinear operator boundedness.

## Key findings

- Established local and global well-posedness in modulation spaces.
- Proved well-posedness for equations with harmonic potentials.
- Demonstrated boundedness of multilinear operators on modulation spaces.

## Abstract

We establish both a local and a global well-posedness theories for the nonlinear Hartree-Fock equations and its reduced analog in the setting of the modulation spaces on $\mathbb R^d$. In addition, we prove similar results when a harmonic potential is added to the equations. In the process, we prove the boundedeness of certain multilinear operators on products of the modulation spaces which may be of independent interest.