Characterization of boundedness of some commutators of fractional maximal functions in terms of $p$-adic vector spaces

This paper gives some characterizations of the boundedness of the maximal or nonlinear commutator of the $p$-adic fractional maximal operator ${\mathcal{M}}_{\alpha }^p$ with the symbols belong to the $p$-adic BMO spaces on (variable) Lebesgue spaces and Morrey spaces over $p$-adic field, by which some new characterizations of BMO functions are obtained in the $p$-adic field context. Meanwhile, Some equivalent relations between the $p$-adic BMO norm and the $p$-adic (variable) Lebesgue or Morrey norm are given.