Local well-posedness for dispersion generalized Benjamin-Ono equations in Fourier-Lebesgue spaces

We prove that the Cauchy problem for the dispersion generalized Benjamin-Ono equation where $0<\alpha \le 1$ \begin{eqnarray*} \left{ \begin{array}{l} \partial_t u+|\partial_x|^{1+α}\partial_x u+uu_x=0,\ u(x,0)=u_0(x), \end{array} \right. \end{eqnarray*} is locally well-posed in the Fourier-Lebesgue space ${\widehat{H}}_r^s(\mathbb{R})$. This is proved via Picard iteration arguments using $X^{s,b}$-type space adapted to the Fourier-Lebesgue space, inspired by the work of Grünrock and Vega. Note that, previously, Molinet, Saut and Tzvetkov \cite{MST2001} proved that the solution map is not $C^2$ in $H^s$ for any $s$ if $0\le \alpha <1$. However, in the Fourier-Lebesgue space, we have a stronger smoothing effect to handle the $high\times low$ interactions.