Lq-ESTIMATES OF MAXIMAL OPERATORS ON THE p-ADIC VECTOR SPACE

Title & Authors
Lq-ESTIMATES OF MAXIMAL OPERATORS ON THE p-ADIC VECTOR SPACE
Kim, Yong-Cheol;

Abstract
For a prime number p, let $\small{\mathbb{Q}_p}$ denote the p-adic field and let $\small{\mathbb{Q}_p^d}$ denote a vector space over $\small{\mathbb{Q}_p}$ which consists of all d-tuples of $\small{\mathbb{Q}_p}$. For a function f $\small{{\in}L_{loc}^1(\mathbb{Q}_p^d)}$, we define the Hardy-Littlewood maximal function of f on $\small{\mathbb{Q}_p^d}$ by $\small{M_pf(x)=sup\frac{1}{\gamma{\in}\mathbb{Z}|B_{\gamma}(x)|H}{\int}_{B\gamma(x)}|f(y)|dy}$, where |E|$\small{_H}$ denotes the Haar measure of a measurable subset E of $\small{\mathbb{Q}_p^d}$ and $\small{B_\gamma(x)}$ denotes the p-adic ball with center x $\small{{\in}\;\mathbb{Q}_p^d}$ and radius $\small{p^\gamma}$. If 1 < q $\small{\leq\;\infty}$, then we prove that $\small{M_p}$ is a bounded operator of $\small{L^q(\mathbb{Q}_p^d)}$ into $\small{L^q(\mathbb{Q}_p^d)}$; moreover, $\small{M_p}$ is of weak type (1, 1) on $\small{L^1(\mathbb{Q}_p^d)}$, that is to say, |{$\small{x{\in}\mathbb{Q}_p^d:|M_pf(x)|}$>$\small{\lambda}$}|$\small{_H{\leq}\frac{p^d}{\lambda}||f||_{L^1(\mathbb{Q}_p^d)},\;\lambda}$ > 0 for any f $\small{{\in}L^1(\mathbb{Q}_p^d)}$.
Keywords
p-adic vector space;the Hardy-Littlewood maximal function;
Language
English
Cited by
1.
Carleson measures and the BMO space on thep-adic vector space, Mathematische Nachrichten, 2009, 282, 9, 1278
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