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 $\lambda \ge 0$, 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 Hölder 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.