Random martingales and localization of maximal inequalities

Let (X, d,μ) be a metric measure space. For ∅ =R ⊆ (0,∞) consider the Hardy–Littlewood maximal operator MRf (x) def = sup r∈R 1 μ(B(x, r)) B(x,r) |f |dμ. We show that if there is an n > 1 such that one has the “microdoubling condition” μ(B(x, (1 + 1 n )r)) μ(B(x, r)) for all x ∈ X andr >0, then the weak (1, 1) norm ofMR has the following localization property: MR L1(X)→L1,∞(X) sup r>0 MR∩[r,nr] L1(X)→L1,∞(X). An immediate consequence is that if (X, d,μ) is Ahlfors–David n-regular then the weak (1, 1) norm ofMR is n log n, generalizing a result of Stein and Strömberg (1983) [47]. We show that this bound is sharp, by constructing a metric measure space (X, d,μ) that is Ahlfors–David n-regular, for which the weak (1, 1) norm of M(0,∞) is n log n. The localization property of MR is proved by assigning to each f ∈ L1(X) a distribution over random martingales for which the associated (random) Doob maximal inequality controls the weak (1, 1) inequality for MR. © 2009 Elsevier Inc. All rights reserved.