On the analytic Birkhoff normal form of the Benjamin–Ono equation and applications

In this paper we prove that the Benjamin–Ono equation admits an analytic Birkhoff normal form in open neighborhood of zero H0s(T,R) for any s>−1/2 where denotes subspace Sobolev space Hs(T,R) elements with mean 0. As application show −1/2<s<0, flow map S0t:H0s(T,R)→H0s(T,R) is nowhere locally uniformly continuous a H0s(T,R).