New Characterization of $$\boldsymbol{p}$$-Adic Herz Spaces with Applications to the Vector-valued Maximal Inequality of Fefferman–Stein Type

Recently, Fu, Wu, and Lu defined $$p$$ -adic Herz spaces. The main goal of this paper is to give a simple proof of the vector-valued maximal inequality of Fefferman–Stein type for $$p$$ -adic Herz spaces. The main ingredient is the new norm equivalence that is adapted to the Muckenhoupt class for the $$n$$ -dimensional $$p$$ -adic space. A review of the $$n$$ -dimensional $$p$$ -adic space is given after the main theorem is stated. This review covers the aspect of measure theory over the $$n$$ -dimensional $$p$$ -adic space. After the proof of this main result, some other possibilities for extensions are discussed. This includes i) weak function spaces together with an application, ii) Hardy operators and iii) variable exponents. Since the theory of weights on the class of variable Lebesgue spaces is missing, the investigation of variable exponents is left for future works. The method used in this paper is simple and promises applications to various situations.