On 1D mass subcritical nonlinear Schr\''odinger and Hartree equations in modulation spaces $M^{p, p'} \ (p<2)$

We establish well-posedness theory for the 1D mass-subcritical nonlinear Schr\"odinger equation (NLS) having power-type nonlinearity $|u|^{\alpha-1}u$ in a certain modulation spaces $M^{p,p'}(\mathbb{R}),$ where $p'$ is a H\"older conjugate of $p$, with $4/3<p<2$ and $p$ sufficiently close to $2$. Modulation spaces have been successfully applied in understanding the dynamics of NLS near the Sobolev scaling critical regularity. In fact, despite cubic NLS is ill-posed in $H^s$ for $s<-1/2$, our analysis reveals that it experiences well-posedness in modulation spaces for a Cauchy data in $(H^{s} \setminus L^{2}) \cap M^{p,p'}$. The proof adopts two different approaches to establish local well-posedness for $\alpha \in (1,5)$, one exploits generalised Strichartz estimates in Fourier-Lebesgue and Lebesgue spaces; the other implements Bourgain's high-low decomposition (BHLD) method in the modulation space setting. The local solution via the (BHLD) method can be extended to global-in-time, but with a certain loss of regularity. We could combine these effectively and establish global well-posedness in $M^{p,p'}$ with the persistence of regularity for $1<\alpha \leq 10/3$. This is the first global result in $M^{p,p'}$ which establishes the persistence of regularity. Similar results are also established for the Hartree equations.