Dispersive decay for the mass-critical Sch\"odinger equation when $d\ge 3$

In this paper we establish the pointwise-in-time dispersive decay for solutions to the mass-critical nonlinear Schr"odinger equation in spatial dimensions $d\ge 3$. Our argument relies on a delicate decomposition of the nonlinearity and an improved linear estimate, which together enable us to control the nonlinear contribution. This work unifies a framework for extending the foundational results established in an earlier paper by Fan, Killip, Visan, and Zhao [Math. Z. $\mathbf{\text{311}}$(1), Paper No. 21, 16 pp (2025)], where the same problem was addressed for spatial dimensions $d=1,2,3$, to the general higher-dimensional setting $d\ge 3$.