Lq-estimates of maximal operators on the p-adic vector space

Abstract

For a prime number p, let Q{double-struck}p denote the p-adic field and let Q{double-struck}dp denote a vector space over Q{double-struck}p which consists of all d-tuples of Q{double-struck}p. For a function f ∈ L1loc(Q{double-struck}dp), we define the Hardy-Littlewood maximal function of f on Q{double-struck}dp by where |E|H denotes the Haar measure of a measurable subset E of Q{double-struck}dp and Bγ(x) denotes the p-adic ball with center x ∈ Q{double-struck}dp and radius pγ. If 1 < q ≤ ∞, then we prove that M{script}p is a bounded operator of Lq(Q{double-struck}dp) into Lq(Q{double-struck}dp); moreover, M{script}p is of weak type (1, 1) on L1(Qdp), that is to say, for any f ∈ L1(Q{double-struck}dp). © 2009 The Korean Mathematical Society.