Characterization of Lipschitz Spaces via Commutators of Fractional Maximal Function on the $p$-Adic Variable Exponent Lebesgue Spaces

Abstract

In this paper, the main aim is to give 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 Lipschitz spaces in the context of the $p$-adic version of variable Lebesgue spaces, by which some new characterizations of the Lipschitz spaces and nonnegative Lipschitz functions are obtained in the $p$-adic field context. Meanwhile, Some equivalent relations between the $p$-adic Lipschitz norm and the $p$-adic variable Lebesgue norm are also given.