  We study the low regularity well-posedness of the 1-dimensional cubic
nonlinear fractional Schr\"odinger equations with L\'{e}vy indices $1 < \alpha
< 2$. We consider both non-periodic and periodic cases, and prove that the
Cauchy problems are locally well-posed in $H^s$ for $s \geq \frac {2-\alpha}4$.
This is shown via a trilinear estimate in Bourgain's $X^{s,b}$ space. We also
show that non-periodic equations are ill-posed in $H^s$ for $\frac {2 -
3\alpha}{4(\alpha + 1)} < s < \frac {2-\alpha}4$ in the sense that the flow map
is not locally uniformly continuous.
