  We show the existence of weak solutions in the extended sense of the Cauchy
problem for the cubic fourth order nonlinear Schr\"odinger equation with
initial data $u_{0}\in X$, where $X\in\{M_{2,q}^{s}(\mathbb R),
H^{\sigma}(\mathbb T), H^{s_{1}}(\mathbb R)+H^{s_{2}}(\mathbb T)\}$ and
$q\in[1,2]$, $s\geq0$, or $\sigma\geq0$, or $s_{2}\geq s_{1}\geq0$. Moreover,
if $M_{2,q}^{s}(\mathbb R)\hookrightarrow L^{3}(\mathbb R)$, or if
$\sigma\geq\frac16$ or if $s_{1}\geq\frac16$ and $s_{2}>\frac12$ we show that
the Cauchy problem is unconditionally wellposed in $X$. Similar results hold
true for all higher order nonlinear Schr\"odinger equations and mixed order NLS
due to a factorization property of the corresponding phase factors. For the
proof we employ the normal form reduction via the differentiation by parts
technique and build upon our previous work
