Global Cauchy problems for the nonlocal (derivative) NLS in $E_{\sigma }^s$

We consider the Cauchy problem for the (derivative) nonlocal NLS in super-critical function spaces $E_{\sigma }^s$ for which the norms are defined by $$\mathrm{\parallel }f{\mathrm{\parallel }}_{E_{\sigma }^s}=\mathrm{\parallel }⟨\xi {⟩}^{\sigma }{2}^{s\mathrm{\mid }\xi \mathrm{\mid }}\widehat{f}(\xi ){\mathrm{\parallel }}_{L^2},s<0,\sigma \in \mathbb{R}\mathrm{.}$$ Any Sobolev space $H^r$ is a subspace of $E_{\sigma }^s$, i.e., $H^r\subset E_{\sigma }^s$ for any $r,\sigma \in \mathbb{R}$ and $s<0$. Let $s<0$ and $\sigma >-1\mathrm{/}2$ ($\sigma >0$) for the nonlocal NLS (for the nonlocal derivative NLS). We show the global existence and uniqueness of the solutions if the initial data belong to $E_{\sigma }^s$ and their Fourier transforms are supported in $(0,\mathrm{\infty })$, the smallness conditions on the initial data in $E_{\sigma }^s$ are not required for the global solutions.