  We consider the low regularity behavior of the fourth order cubic nonlinear
Schr\"odinger equation (4NLS) \begin{align*} \begin{cases}
i\partial_tu+\partial_x^4u=\pm \vert u \vert^2u, \quad(t,x)\in \mathbb{R}\times
\mathbb{R}\\ u(x,0)=u_0(x)\in H^s\left(\mathbb{R}\right). \end{cases}
\end{align*} In arXiv:1911.03253, the author showed that this equation is
globally well-posed in $H^s, s\geq -\frac{1}{2}$ and ill-posedness in the sense
that the solution map fails to be uniformly continuous for
$-\frac{15}{14}<s<-\frac{1}{2}$. Therefore, $s=-\frac{1}{2}$ is the lowest
regularity that can be handled by the contraction argument. In spite of this
ill-posedness result, we obtain a priori bound below $s<-1/2$. This a priori
estimate guarantees the existence of a weak solution for $-3/4<s<-1/2$. But we
cannot establish full well-posedness because of the lack of energy estimate of
differences of solutions. Our method is inspired by Koch-Tataru \cite{KT2007}.
We use the $U^p$ and $V^p$ based spaces adapted to frequency dependent time
intervals on which the nonlinear evolution can be still described by linear
dynamics.
