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We shall deduce some special regularity properties of solutions to the IVPassociated to the Benjamin-Ono equation. Mainly, for datum $u_0\inH^{3/2}(\mathbb R)$ whose restriction belongs to $H^m((b,\infty))$ for some$m\in\mathbb Z^+,\,m\geq 2,$ and $b\in \mathbb R$ we shall prove that therestriction of the corresponding solution $u(\cdot,t)$ belongs to$H^m((\beta,\infty))$ 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 astime evolves.

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