  In this work we study some regularity properties associated to the initial
value problem (IVP) \begin{equation}\label{main1} \left\{ \begin{array}{ll}
\partial_{t}u-\partial_{x_{1}}(-\Delta)^{\alpha/2} u+u\partial_{x_{1}}u=0,
\quad 0< \alpha\leq 2,& \\ u(x,0)=u_{0}(x),\quad x=(x_{1},x_{2},\dots,x_{n})\in
\mathbb{R}^{n},\, n\geq 2,\quad t\in\mathbb{R},& \\ \end{array} \right.
\end{equation} where $(-\Delta)^{\alpha/2}$ denotes the $n-$dimensional
fractional Laplacian.
  We show that solutions to the IVP (0.1) with initial data in a suitable
Sobolev space exhibit a local smoothing effect in the spatial variable of
$\frac{\alpha}{2}$ derivatives, almost everywhere in time. One of the main
difficulties that emerge when trying to obtain this regularizing effect
underlies that the operator in consideration is non-local, and the property we
are trying to describe is local, so new ideas are required. Nevertheless, to
avoid these problems, we use a perturbation argument replacing
$(-\Delta)^{\frac{\alpha}{2}}$ by $(I-\Delta)^{\frac{\alpha}{2}},$ that through
the use of pseudo-differential calculus allows us to show that solutions become
locally smoother by $\frac{\alpha}{2}$ of a derivative in all spatial
directions.
  As a by-product, we use this particular smoothing effect to show that the
extra regularity of the initial data on some distinguished subsets of the
Euclidean space is propagated by the flow solution with infinity speed.
