Communications of the
Korean Mathematical Society CKMS

For a prime number $p$, let ${\mathbb Q}_p$ denote the $p$-adic field and let ${\mathbb Q}_p^d$ denote a vector space over ${\mathbb Q}_p$ which consists of all $d$-tuples of ${\mathbb Q}_p$. For a function $f\in L^1_{loc}({\mathbb Q}_p^d)$, we define the Hardy-Littlewood maximal function of $f$ on ${\mathbb Q}_p^d$ by $${\mathcal M}_p\,f(\bold x)=\sup_{\gamma\in {\mathbb Z}}\frac{1}{|B_{\gamma}(\bold x)|_H}\int_{B_{\gamma}(\bold x)} |f(\bold y)|\,d{\bold y},$$ where $|E|_H$ denotes the Haar measure of a measurable subset $E$ of ${\mathbb Q}_p^d$ and $B_{\gamma}(\bold x)$ denotes the $p$-adic ball with center $\bold x\in {\mathbb Q}_p^d$ and radius $p^{\gamma}$. If $\,1\lambda\}|_H\le\frac{p^d}{\lambda} \,\|f\|_{L^1({\mathbb Q}_p^d)},\,\,\lambda>0$$ for any $f\in L^1({\mathbb Q}_p^d)$.

Keywords: $p$-adic vector space, the Hardy-Littlewood maximal function

MSC numbers: 11S80, 11K70, 11E9