  Consider the mass-critical nonlinear Schr\"odinger equations in both focusing
and defocusing cases for initial data in $L^2$ in space dimension N. By
Strichartz inequality, solutions to the corresponding linear problem belong to
a global $L^p$ space in the time and space variables, where $p=2+4/N$. In 1D
and 2D, the best constant for the Strichartz inequality was computed by
D.~Foschi who has also shown that the maximizers are the solutions with
Gaussian initial data.
  Solutions to the nonlinear problem with small initial data in $L^2$ are
globally defined and belong to the same global $L^p$ space. In this work we
show that the maximum of the $L^p$ norm is attained for a given small mass. In
addition, in 1D and 2D, we show that the maximizer is unique and obtain a
precise estimate of the maximum. In order to prove this we show that the
maximum for the linear problem in 1D and 2D is nondegenerated.
