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We consider the Cauchy problem for the (derivative) nonlocal NLS insuper-critical function spaces $E^s_\sigma$ for which the norms are defined by$$ \|f\|_{E^s_\sigma} = \|\langle\xi\rangle^\sigma2^{s|\xi|}\hat{f}(\xi)\|_{L^2}, \ s<0, \ \sigma \in \mathbb{R}. $$ Any Sobolevspace $H^{r}$ is a subspace of $E^s_\sigma$, i.e., $H^r \subset E^s_\sigma$ forany $ r,\sigma \in \mathbb{R}$ and $s<0$. Let $s<0$ and $\sigma>-1/2$ ($\sigma>0$) for the nonlocal NLS (for the nonlocal derivative NLS). We show the globalexistence and uniqueness of the solutions if the initial data belong to$E^s_\sigma$ and their Fourier transforms are supported in $(0, \infty)$, thesmallness conditions on the initial data in $E^s_\sigma$ are not required forthe global solutions.

Global Cauchy problems for the nonlocal (derivative) NLS in $E^s_\sigma$