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 ${\mathrm{\partial }}_{t}u+\mathcal{H}{\mathrm{\partial }}_x^{2}u\pm u^k{\mathrm{\partial }}_{x}u=0$, $k\ge 4$ are locally well-posed in the scaling invariant spaces ${\dot{H}}^{s_k}(\mathbb{R})$ where $s_k=1\mathrm{/}2-1\mathrm{/}k$. Our results also hold in the non-homogeneous spaces $H^{s_k}(\mathbb{R})$. In the case $k=3$, local well-posedness is obtained in $H^s(\mathbb{R})$, $s>1\mathrm{/}3$.