Tame majorant analyticity for the Birkhoff map of the defocusing Nonlinear Schr\"odinger equation on the circle

For the defocusing Nonlinear Schr"odinger equation on the circle, we construct a Birkhoff map $\mathrm{\Phi }$ which is tame majorant analytic in a neighborhood of the origin. Roughly speaking, majorant analytic means that replacing the coefficients of the Taylor expansion of $\mathrm{\Phi }$ by their absolute values gives rise to a series (the majorant map) which is uniformly and absolutely convergent, at least in a small neighborhood. Tame majorant analytic means that the majorant map of $\mathrm{\Phi }$ fulfills tame estimates. The proof is based on a new tame version of the Kuksin-Perelman theorem, which is an infinite dimensional Vey type theorem.