On the Cauchy problem of a two-dimesional Benjamin-Ono equation

In this work we shall show that the Cauchy problem \begin{equation} \left\{ \begin{aligned} &(u_t+u^pu_x+\mathcal H\partial_x^2u+ \alpha\mathcal H\partial_y^2u )_x - \gamma u_{yy}=0 \quad p\in{\nat} &u(0;x,y)=\phi{(x,y)} \end{aligned} \right. \end{equation} is locally well-posed in the Sobolev spaces \(H^s({\re}^2)\), \(X^s\) and weighted spaces \(X_s(w^2)\), for \(s>2\).