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

Abstract: 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.