  In this paper, we study the wellposedeness of the Hall-magnetohydrodynamic
system augmented by the effect of electron inertia. Our main result consists of
generalising the wellposedness one in \cite{Zhao} from the Sobolev context to
the general Besov spaces and Kato-Herz space, then we show that we can reduce
the requared regularity of the magnetic field in the first result modulo an
additional condition on the maximal time of existence. Finally, we show that
the $\widehat{L}^p$ (and eventually the $L^p$) norm of the solution
$(u,B,\nabla \times B)$ associated to an initial data in $
\widehat{B}^{\frac{3}{p}-1}_{p,\infty}(\mathbb{R}^3)$, is controled by
$t^{-\frac{1}{2}(1-\frac{3}{p})}$, for all $p\in (3,\infty)$, which provides a
polynomial decay to zero of the $\widehat{L}^p$ norm of the solution.
