  In this work we consider the Cauchy problem for the cubic Schr\"odinger
equation posed on cylinder $\mathbb{R}\times\mathbb{T}$ with fractional
derivatives $(-\partial_y^2)^{\alpha},\, \alpha >0$, in the periodic direction.
The spatial operator includes elliptic and hyperbolic regimes. We prove $L^2$
global well-posedness results when $\alpha > 1$ by proving a $L^4 - L^2$
Strichartz inequality for the linear equation, following the ideas developep by
H. Takaoka and N. Tzvetkov in the the case of the Laplacian operator. Further,
these results remain valid on the euclidean environment $\mathbb{R}^2$, so
well-posedness in $L^2$ are also achieved in this case. Our proof in the
elliptic (hyperbolic) case does not work in the case $0<\alpha <1$ ($0<\alpha
\leq 1$), respectively.
