Well-posedness results for the 3D Zakharov-Kuznetsov equation

We prove the local well-posedness of the three-dimensional Zakharov-Kuznetsov equation ${\mathrm{\partial }}_{t}u+\mathrm{\Delta }{\mathrm{\partial }}_{x}u+u{\mathrm{\partial }}_{x}u=0$ in the Sobolev spaces $H^s({\mathbb{R}}^3)$, $s>1$, as well as in the Besov space $B_2^{1,1}({\mathbb{R}}^3)$. The proof is based on a sharp maximal function estimate in time-weighted spaces.