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<α ≤ 1 {eqnarray*} \{ {array}{l} ∂_t u+|∂_x|1+α∂_x u+uu_x=0,\\ u(x,0)=u_0(x), {array} . {eqnarray*} is C² locally well-posed in the Fourier-Lebesgue space Hˢᵣ(R). This is proved via Picard iteration arguments using Xs,b-type space adapted to the Fourier-Lebesgue space, inspired by the work of Gr\"unrock and Vega. We also proved the ill-posedness result that shows the range is sharp. Note that, previously, Molinet, Saut and Tzvetkov {MST2001} proved that the solution map is not C² in Hˢ for any s if 0≤ α<1. However, in the Fourier-Lebesgue space, we have a stronger smoothing effect to handle the high× low interactions.