Necessary and sufficient conditions for boundedness of commutators of maximal function on the $p$-adic vector spaces

In this paper, we first show that the $p$-adic version of maximal function ${\mathcal{M}}_{L\log L}^p$ is equivalent to the maximal function ${\mathcal{M}}^p({\mathcal{M}}^p)$ and that the class of functions for which the maximal commutators and the commutator with the $p$-adic version of maximal function or the maximal sharp function are bounded on the $p$-adic vector spaces are characterized and proved to be the same. Moreover, new pointwise estimates for these operators are proved.