  In this paper we prove a wellposedness result of the KdV equation on the
space of periodic pseudo-measures, also referred to as the Fourier Lebesgue
space $\mathscr{F}\ell^{\infty}(\mathbb{T},\mathbb{R})$, where
$\mathscr{F}\ell^{\infty}(\mathbb{T},\mathbb{R})$ is endowed with the weak*
topology. Actually, it holds on any weighted Fourier Lebesgue space
$\mathscr{F}\ell^{s,\infty}(\mathbb{T},\mathbb{R})$ with $-1/2 < s \le 0$ and
improves on a wellposedness result of Bourgain for small Borel measures as
initial data. A key ingredient of the proof is a characterization for a
distribution $q$ in the Sobolev space $H^{-1}(\mathbb{T},\mathbb{R})$ to be in
$\mathscr{F}\ell^{\infty}(\mathbb{T},\mathbb{R})$ in terms of asymptotic
behavior of spectral quantities of the Hill operator $-\partial_{x}^{2} + q$.
In addition, wellposedness results for the KdV equation on the Wiener algebra
are proved.
