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 ∈ H 3/4+ (ℝ) whose restriction belongs to H l ((b, ∞)) for some l ∈ ℤ+ and b ∈ ℝ we prove that the restriction of the corresponding solution u(·, t) belongs to H l ((β, ∞)) for any β ∈ ℝ and any t ∈ (0, T). Thus, this type of regularity propagates with infinite speed to its left as time evolves.