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

Abstract

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.

Keywords

• $ p $-adic vector space
• maximal function
• commutator
• bmo space
• morrey spaces