  We study special regularity and decay properties of solutions to the IVP
associated to the $k$-generalized KdV equations. In particular, for datum
$u_0\in H^{3/4^+}(\mathbb R)$ whose restriction belongs to $H^l((b,\infty))$
for some $l\in\mathbb Z^+$ and $b\in \mathbb R$ we prove that the restriction
of the corresponding solution $u(\cdot,t)$ belongs to $H^l((\beta,\infty))$ for
any $\beta \in \mathbb R$ and any $t\in (0,T)$. Thus, this type of regularity
propagates with infinite speed to its left as time evolves.
