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In this work we study some regularity properties associated to the initialvalue 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 $n-$dimensionalfractional Laplacian.  We show that solutions to the IVP (0.1) with initial data in a suitableSobolev space exhibit a local smoothing effect in the spatial variable of$\frac{\alpha}{2}$ derivatives, almost everywhere in time. One of the maindifficulties that emerge when trying to obtain this regularizing effectunderlies that the operator in consideration is non-local, and the property weare trying to describe is local, so new ideas are required. Nevertheless, toavoid these problems, we use a perturbation argument replacing$(-\Delta)^{\frac{\alpha}{2}}$ by $(I-\Delta)^{\frac{\alpha}{2}},$ that throughthe use of pseudo-differential calculus allows us to show that solutions becomelocally smoother by $\frac{\alpha}{2}$ of a derivative in all spatialdirections.  As a by-product, we use this particular smoothing effect to show that theextra regularity of the initial data on some distinguished subsets of theEuclidean space is propagated by the flow solution with infinity speed.

On Kato's smoothing effect for a fractional version of the Zakharov-Kuznetsov equation