Maximizers for the Strichartz norm for small solutions of mass-critical NLS

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Consider the mass-critical nonlinear Schrödinger equations in both focusing and defocusing cases for initial data in L² in space dimension N. By Strichartz inequality, solutions to the corresponding linear problem belong to a global Lᵖ 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 L² are globally defined and belong to the same global Lᵖ space. In this work we show that the maximum of the Lᵖ 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.