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 pseudomeasures, also referred to as the Fourier Lebesgue space Fℓ∞(T,R), where Fℓ∞(T,R) is endowed with the weak* topology. Actually, it holds on any weighted Fourier Lebesgue space Fℓs,∞(T,R) with −1/2<s≤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(T,R) to be in Fℓ∞(T,R) in terms of asymptotic behavior of spectral quantities of the Hill operator −∂2x+q. In addition, wellposedness results for the KdV equation on the Wiener algebra are proved.