Vacuum initial data with minimal decay and borderline decay

In this note, we show that the conical solution-operator method of Mao-Tao in [Localized initial data for Einstein equations] applies to a simple construction of vacuum asymptotically flat initial data at minimal and borderline decay thresholds, corresponding to the global and exterior stability of Minkowski spacetime proved by the first named author in [Global stability of Minkowski spacetime with minimal decay] and [Exterior stability of Minkowski spacetime with borderline decay].