On the propagation of regularities in solutions of the Benjamin-Ono equation

We shall deduce some special regularity properties of solutions to the IVP associated to the Benjamin-Ono equation. Mainly, for datum $u_0\in H^{3/2}(\mathbb R)$whoserestrictionbelongsto$H^m((b,\infty))$forsome$m\in\mathbb Z^+,,m\geq 2,$and$b\in \mathbb R$ we shall prove that the restriction of the corresponding solution $u(\cdot ,t)$ belongs to $H^m((\beta ,\mathrm{\infty }))$ for any $\beta \in \mathbb{R}$ and any $t>0$. Therefore, this type of regularity of the datum travels with infinite speed to its left as time evolves.