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

Abstract

<abstract><p>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.</p></abstract>

$$L\log L$$ Type Estimates for Commutators of Fractional Integral Operators on the p-Adic Vector Space

“…In [10], the equivalence of M p (M p ) and M p L log L is given by the following which is very analogous to Euclidean setting. Lemma 2.3 Let f be a locally integrable function on Q n p .…”

“…In order to estimate I, from the proof of the formula (2.5) in [10], we see that for any g with supp g ⊂ B γ (x)…”

Endpoint Estimates for Commutators of Fractional Integral Operators on the p-Adic Vector Space

“…In [11], the size of M p (M p ) is given by the following which is similar to Euclidean setting. Lemma 2.3 Let f be a locally integrable function on Q n p .…”

“…In order to estimate I, from the proof of the formula (2.5) in [11], we see that for any g with supp g ⊂ B γ (x)…”