Global Well-posedness for the fourth order nonlinear Schr\"{o}dinger equations with small rough data in high demension

For $n\ge 2$, we establish the smooth effects for the solutions of the linear fourth order Shr"{o}dinger equation in anisotropic Lebesgue spaces with ${\mathrm{\square }}_k$-decomposition. Using these estimates, we study the Cauchy problem for the fourth order nonlinear Schr"{o}dinger equations with three order derivatives and obtain the global well posedness for this problem with small data in modulation space $M_{2,1}^{9\mathrm{/}2}({\text{Real}}^n)$.