Well-posedness for the generalized Benjamin-Ono equations with arbitrary large initial data in the critical space

We prove that the generalized Benjamin-Ono equations $\partial_tu+\mathcal{H}\partial_x^2u\pm u^k\partial_xu=0$, $k\geq 4$ are locally well-posed in the scaling invariant spaces $\dot{H}^{s_k}(\R)$ where $s_k=1/2-1/k$. Our results also hold in the non-homogeneous spaces $H^{s_k}(\R)$. In the case $k=3$, local well-posedness is obtained in $H^{s}(\R)$, $s>1/3$.