Local Well-posedness for dispersion generalized Benjamin-Ono equations in Sobolev spaces

We prove that the Cauchy problem for the dispersion generalized Benjamin-Ono equation $$\partial_t u+|\partial_x|^{1+\alpha}\partial_x u+uu_x=0,\ u(x,0)=u_0(x),$$ is locally well-posed in the Sobolev spaces $H^s$ for $s>1-\alpha$ if $0\leq \alpha \leq 1$. The new ingredient is that we develop the methods of Ionescu, Kenig and Tataru \cite{IKT} to approach the problem in a less perturbative way, in spite of the ill-posedness results of Molinet, Saut and Tzvetkovin \cite{MST}. Moreover, as a bi-product we prove that if $0<\alpha \leq 1$ the corresponding modified equation (with the nonlinearity $\pm uuu_x$) is locally well-posed in $H^s$ for $s\geq 1/2-\alpha/4$.