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

For a prime number p, let <TEX>Qₚ</TEX> denote the p-adic field and let <TEX>Qₚᵈ</TEX> denote a vector space over <TEX>Qₚ</TEX> which consists of all d-tuples of <TEX>Qₚ</TEX>. For a function f <TEX>∈Lloc¹(Qₚᵈ)</TEX>, we define the Hardy-Littlewood maximal function of f on <TEX>Qₚᵈ</TEX> by <TEX>Mₚf(x)=sup1γ∈Z|Bγ(x)|H∫Bγ(x)|f(y)|dy</TEX>, where |E|<TEX>H</TEX> denotes the Haar measure of a measurable subset E of <TEX>Qₚᵈ</TEX> and <TEX>B_γ(x)</TEX> denotes the p-adic ball with center x <TEX>∈\;Qₚᵈ</TEX> and radius <TEX>p^γ</TEX>. If 1 < q <TEX>≤\;∞</TEX>, then we prove that <TEX>Mₚ</TEX> is a bounded operator of <TEX>Lq(Qₚᵈ)</TEX> into <TEX>Lq(Qₚᵈ)</TEX>; moreover, <TEX>Mₚ</TEX> is of weak type (1, 1) on <TEX>L¹(Qₚᵈ)</TEX>, that is to say, |{<TEX>x∈Qₚᵈ:|Mₚf(x)|</TEX>><TEX>λ</TEX>}|<TEX>H≤pᵈ/λ||f||L¹(Qₚᵈ),\;λ</TEX> > 0 for any f <TEX>∈L¹(Qₚᵈ)</TEX>.