Remark on the local well-posedness for NLS with the modulated dispersion

We consider the Cauchy problem of the nonlinear Schr\"odinger equation with the modulated dispersion and power type nonlinearities in any spatial dimensions. We adapt the Young integral theory developed by Chouk-Gubinelli [K. Chouk and M, Gubinelli, Comm. Partial Differential Equations 40 (2015)] and multilinear estimates which are based on divisor counting, and show the local well-posedness. Our result generalizes the result by Chouk-Gubinelli.