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We prove that the generalized Benjamin-Ono equations$\partial_tu+\mathcal{H}\partial_x^2u\pm u^k\partial_xu=0$, $k\geq 4$ arelocally 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$H^{s}(\R)$, $s>1/3$.

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