Cauchy problem for the Ostrovsky equation in spaces of low regularity

In this article we consider the initial value problem for the Ostrovsky equation:∂tu−∂x3u∓∂x−1u+u∂xu=0,x∈R,t∈R,u(x,0)=u0(x), with initial data in Sobolev spaces Hs(R). Using Bourgain spaces, we prove that the problem is locally well-posed for s>−12 if the sign of the third term of the equation is “−” and for s>−34 if the sign of this term is “+”.