| Abstract: | Motivated by perturbation theory, we prove that the nonlinear part ({H^{*}}) of the KdV Hamiltonian ({H^{kdv}}), when expressed in action variables ({I = (I_{n})_{n geqslant 1}}), extends to a real analytic function on the positive quadrant ({ell^{2}_{+}(mathbb{N})}) of ({ell^{2}(mathbb{N})}) and is strictly concave near ({0}). As a consequence, the differential of ({H^{*}}) defines a local diffeomorphism near 0 of ({ell_{mathbb{C}}^{2}(mathbb{N})}). Furthermore, we prove that the Fourier-Lebesgue spaces ({mathcal{F}mathcal{L}^{s,p}}) with ({-1/2 leqslant s leqslant 0}) and ({2 leqslant p < infty}), admit global KdV-Birkhoff coordinates. In particular, it means that ({ell^{2}_+(mathbb{N})}) is the space of action variables of the underlying phase space ({mathcal{F}mathcal{L}^{-1/2,4}}) and that the KdV equation is globally in time ({C^{0}})-well-posed on ({mathcal{F}mathcal{L}^{-1/2,4}}). |
