On the unconditional uniqueness for NLS in $H^s$

In this article, we study the unconditional uniqueness of $\dot H^s$, $0<s< 1$, solutions for the nonlinear Schr\"odinger equation $i\partial_t u +\Delta u+ c |u|^\alpha u=0$ in ${\mathbb R}^n$. We give a unified proof of the previously known results in the subcritical cases and critical cases, and we also extend these results to some previously unsettled cases. Our proof uses in particular negative order Sobolev spaces (or Besov spaces), general Strichartz estimates, and the improved regularity property for the difference of two solutions.