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 $0leq alpha leq 1$. 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 $sgeq 1/2-alpha/4$.