On well-posedness for the Benjamin-Ono equation

We prove existence of solutions for the Benjamin-Ono equation with data in $H^s(\mathbb{R})$, $s>0$. Thanks to conservation laws, this yields global solutions for $H^{\frac{1}{2}}(\mathbb{R})$ data, which is the natural ``finite energy’’ class. Moreover, inconditional uniqueness is obtained in $L_t^{\mathrm{\infty }}(H^{\frac{1}{2}}(\mathbb{R}))$, which includes weak solutions, while for $s>\frac{3}{20}$, uniqueness holds in a natural space which includes the obtained solutions.