  In this paper, we give a characterization of mixed $\lambda$-central bounded
mean oscillation space $\mathrm{CBMO}^{\vec{q},\lambda}(\mathbb{R}^{n})$ via
the boundedness of the commutators of $n$-dimensional Hardy operator
$\mathcal{H}$ and its dual operator $\mathcal{H}^{*}$ on mixed Lebesgue space
$L^{\vec{p}}(\mathbb{R}^{n})$. In addition, we also establish the boundedness
of commutators $\mathcal{H}_{b}$ and $\mathcal{H}^{*}_{b}$ generated with
$\mathrm{CBMO}^{\vec{q},\lambda}(\mathbb{R}^{n})$ function $b$ on mixed
$\lambda$-central Morrey space $\mathcal{B}^{\vec{q},\lambda}(\mathbb{R}^{n})$,
respectively.
