L^q-ESTIMATES OF MAXIMAL OPERATORS ON THE p-ADIC VECTOR SPACE

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 ${\in}L_{loc}^1(\mathbb{Q}_p^d)$, we define the Hardy-Littlewood maximal function of f on $\mathbb{Q}_p^d$ by $$M_pf(x)=sup\frac{1}{\gamma{\in}\mathbb{Z}|B_{\gamma}(x)|H}{\int}_{B\gamma(x)}|f(y)|dy$$, where |E|$_H$ denotes the Haar measure of a measurable subset E of $\mathbb{Q}_p^d$ and $B_\gamma(x)$ denotes the p-adic ball with center x ${\in}\;\mathbb{Q}_p^d$ and radius $p^\gamma$. If 1 < q $\leq\;\infty$, then we prove that $M_p$ is a bounded operator of $L^q(\mathbb{Q}_p^d)$ into $L^q(\mathbb{Q}_p^d)$; moreover, $M_p$ is of weak type (1, 1) on $L^1(\mathbb{Q}_p^d)$, that is to say, |{$x{\in}\mathbb{Q}_p^d:|M_pf(x)|$>$\lambda$}|$_H{\leq}\frac{p^d}{\lambda}||f||_{L^1(\mathbb{Q}_p^d)},\;\lambda$ > 0 for any f ${\in}L^1(\mathbb{Q}_p^d)$.