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

We establish well-posedness theory for the 1D mass-subcritical nonlinear Schrödinger equation (NLS) having power-type nonlinearity $\mathrm{\mid }u{\mathrm{\mid }}^{\alpha -1}u$ in a certain modulation spaces $M^{p,p^{\mathrm{\prime }}}(\mathbb{R}),$ where $p^{\mathrm{\prime }}$ is a Hölder conjugate of $p$, with $4\mathrm{/}3<p<2$ 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 $s<-1\mathrm{/}2$, our analysis reveals that it experiences well-posedness in modulation spaces for a Cauchy data in $(H^s\setminus L^2)\cap M^{p,p^{\mathrm{\prime }}}$. 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 $M^{p,p^{\mathrm{\prime }}}$ with the persistence of regularity for $1<\alpha \le 10\mathrm{/}3$. This is the first global result in $M^{p,p^{\mathrm{\prime }}}$ which establishes the persistence of regularity. Similar results are also established for the Hartree equations.