  This paper is focused on the modified Zakharov-Kusnetsov equation. We prove
the associated Cauchy problem is locally (in time) well-posed in $H^s(\R \times
\T)$ for $s >1$. The new ingredient in this work is a trilinear estimate in the
context of Bourgain spaces which controls the cubic non-linearity of the
modified Zakharov-Kuznetsov equation in the contraction argument.
