L^q-estimates of maximal operators on the p-adic vector space
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For a prime number p, let Q_p denote the p-adic field and let Q^d_p denote a vector space over Q_p which consists of all d-tuples of Q_p. For a function f ∈L^1_loc(Q^d_p), we define the Hardy-Littlewood maximal function of f on Q^d_p by <수식> where |E|_H denotes the Haar measure of a measurable subset E of Q^d_p and B_γ(x) denotes the p-adic ball with center x∈Q^d_p and radius p^γ. If 1 < q ≤ ∞, then we prove that M_p is a bounded operator of L_q(Q^d_p) into Lq(Q^d_p); moreover, M_p is of weak type (1, 1) on L^1(Q^d_p), that is to say, <수식> for any f ∈ L^1(Q^d_p).
Abstract
For a prime number p, let Q_p denote the p-adic field and let Q^d_p denote a vector space over Q_p which consists of all d-tuples of Q_p. For a function f ∈L^1_loc(Q^d_p), we define the Hardy-Littlewood maximal function of f on Q^d_p by <수식> where |E|_H denotes the Haar measure of a measurable subset E of Q^d_p and B_γ(x) denotes the p-adic ball with center x∈Q^d_p and radius p^γ. If 1 < q ≤ ∞, then we prove that M_p is a bounded operator of L_q(Q^d_p) into Lq(Q^d_p); moreover, M_p is of weak type (1, 1) on L^1(Q^d_p), that is to say, <수식> for any f ∈ L^1(Q^d_p).
참고문헌 (8)
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