  We show that the initial value problem associated to the dispersive
generalized Benjamin-Ono-Zakharov-Kuznetsov equation$$ u\_t-D\_x^\alpha u\_{x}
+ u\_{xyy} = uu\_x,\quad (t,x,y)\in\R^3,\quad 1\le \alpha\le 2,$$is locally
well-posed in the spaces $E^s$, $s\textgreater{}\frac 2\alpha-\frac 34$,
endowed with the norm$\|f\|\_{E^s} = \|\langle
|\xi|^\alpha+\mu^2\rangle^s\hat{f}\|\_{L^2(\R^2)}.$As a consequence, we get the
global well-posedness in the energy space $E^{1/2}$ as soon as
$\alpha\textgreater{}\frac 85$. The proof is based on the approach of the short
time Bourgain spaces developed by Ionescu, Kenig and Tataru \cite{IKT} combined
with new Strichartz estimates and a modified energy.
