The Cauchy problem $u_t - |D|^{\alpha}u_x + uu_x=0$ in
$(-T,T) \times \mathbb{R}$, $u(0)=u_0$, is studied for $1<
\alpha <2$. Using suitable spaces of Bourgain type, local
well-posedness for initial data $u_0 \in H^s(\mathbb{R}) \cap
\dot{H}^{-\omega}(\mathbb{R})$ for any $s >
-\tfrac{3}{4}(\alpha-1)$, $\omega:=1/\alpha-1/2$ is
shown. This includes existence, uniqueness, persistence, and
analytic dependence on the initial data. These results are
sharp with respect to the low frequency condition in the sense
that if $\omega<1/\alpha-1/2$, then the flow map is not
$C^2$ due to the counterexamples previously known. By using a
conservation law, these results are extended to global
well-posedness in $H^s(\mathbb{R}) \cap
\dot{H}^{-\omega}(\mathbb{R})$ for $s \geq 0$,
$\omega=1/\alpha-1/2$, and real valued initial data.