Unconditional well-posedness for the nonlinear Schr\"odinger equation in Bessel potential spaces

The Cauchy problem for the nonlinear Schr\"odinger equation is called unconditionally well posed in a data space E if it is well posed in the usual sense and the solution is unique in the space $C([0,T]; E)$. In this paper, this notion of unconditional well-posedness is redefined so that it covers Lᵖ-based Sobolev spaces as data space E and it is equivalent to the usual one when E is an L²-based Sobolev space Hˢ. Next, based on this definition, it is shown that the Cauchy problem for the 1D cubic NLS is unconditionally well posed in Bessel potential spaces Hˢₚ for 4/3<p≤ 2 under certain regularity assumptions on s.