On the wellposedness of the KdV equation on the space of pseudomeasures

Abstract

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.