  In this paper we study some properties of propagation of regularity of
solutions of the dispersive generalized Benjamin-Ono (BO) equation. This model
defines a family of dispersive equations, that can be seen as a dispersive
interpolation between Benjamin-Ono equation and Korteweg-de Vries (KdV)
equation.
  Recently, it has been showed that solutions of the KdV equation and
Benjamin-Ono equation, satisfy the following property: if the initial data has
some prescribed regularity on the right hand side of the real line, then this
regularity is propagated with infinite speed by the flow solution.
  In this case the nonlocal term present in the dispersive generalized
Benjamin-Ono equation is more challenging that the one in BO equation. To deal
with this a new approach is needed. The new ingredient is to combine commutator
expansions into the weighted energy estimate. This allow us to obtain the
property of propagation and explicitly the smoothing effect.
