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

TL;DR

This paper characterizes the boundedness of certain commutators of p-adic fractional maximal operators on variable Lebesgue and Morrey spaces, revealing new insights into p-adic BMO functions and their norms.

Contribution

It provides novel characterizations of p-adic BMO functions via boundedness of commutators of fractional maximal operators on various function spaces.

Findings

New characterizations of p-adic BMO functions.

Equivalent relations between p-adic BMO norm and Lebesgue/Morrey norms.

Boundedness criteria for commutators on p-adic spaces.

Abstract

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.