A characterization of mixed $\lambda$-central $\mathrm{BMO}$ space via the commutators of Hardy type operators

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.