  We prove that the renormalized defocusing mKdV equation on the circle is
locally in time $C^{0}$-wellposed on the Fourier Lebesgue space
${\mathcal{F}\ell}^p$ for any $2 < p < \infty$. The result implies that the
defocusing mKdV equation itself is illposed on these spaces since the
renormalizing phase factor becomes infinite. The proof is based on the fact
that the mKdV equation is an integrable PDE whose Hamiltonian is in the NLS
hierarchy. A key ingredient is a novel way of representing the bi-infinite
sequence of frequencies of the renormalized defocusing mKdV equation, allowing
to analytically extend them to ${\mathcal{F}\ell}^p$ for any $2 \le p < \infty$
and to deduce asymptotics for $n \to \pm \infty$.
