On the Cauchy problem for the Hartree type equation in the Wiener algebra

Abstract

We consider the mass-subcritical Hartree equation with a homogeneous kernel in the space of square integrable functions whose Fourier transform is integrable. We prove a global well-posedness result in this space. On the other hand, we show that the Cauchy problem is not even locally well-posed if we simply work in the space of functions whose Fourier transform is integrable. Similar results are proven when the kernel is not homogeneous and is such that its Fourier transform belongs to some Lebesgue space.