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 \( \mathbb {R}\times \mathbb {T}\) 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.