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

Abstract

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 ${\mathcal{H}}^{\ast }$ on mixed Lebesgue space $L^{\vec{p}}({\mathbb{R}}^n)$. In addition, we also establish the boundedness of commutators ${\mathcal{H}}_b$ and ${\mathcal{H}}_b^{\ast }$ 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.

14 pages