A note on the well-posedness in the energy space for the generalized ZK equation posed on $\mathbb{R}\times\mathbb{T}$

In this note, we prove the local well-posedness in the energy space of the $k$-generalized Zakharov-Kuznetsov equation posed on $\R\times \T$ for any power non-linearity $k\ge 2$. Moreover, we obtain global solutions under a precise smallness assumption on the initial data by proving a sharp Gagliardo Nirenberg type inequality.