The Cauchy problem for a family of two-dimensional fractional Benjamin-Ono equations

In this work we prove that the initial value problem (IVP) associated to the fractional two-dimensional Benjamin-Ono equation $$\begin{array}{rll}{u}_t+D_x^{\alpha }{u}_x+\mathcal{H}{u}_{yy}+u{u}_x& =0,(x,y)\in {\mathbb{R}}^2,t\in \mathbb{R},u(x,y,0)& =u_0(x,y),\end{array}\},$$ where $0<\alpha \le 1$, $D_x^{\alpha }$ denotes the operator defined through the Fourier transform by \begin{align} (D_x^{\alpha}f)\widehat{;}(\xi,\eta):=|\xi|^{\alpha}\widehat{f}(\xi,\eta),, \end{align} and $\mathcal{H}$ denotes the Hilbert transform with respect to the variable $x$, is locally well posed in the Sobolev space $H^s({\mathbb{R}}^2)$ with $s>{\displaystyle \frac{3}{2}}+{\displaystyle \frac{1}{4}}(1-\alpha )$.