Characterizations of the mixed central Campanato spaces via the commutator operators of Hardy type

The purpose of this paper is to establish some characterizations of mixed central Campanato space $\mathfrak{C}^{\vec{p},\lambda}(\mathbb{R}^{n})$, via the boundedness of the commutator operators of Hardy type. Unlike the case $\lambda\geq0$, there are some technical difficulties caused by $\lambda<0$ to be overcome. In addition, an extra assumption called as the mixed version of the reverse H\"{o}lder class be required in the proof of the converse characterization. Moreover, some further interesting conclusions for the Hardy type operators on mixed $\lambda$-central Morrey space $\mathcal{B}^{\vec{p},\lambda}(\mathbb{R}^{n})$ are also derived.