Random martingales and localization of maximal inequalities

Abstract

Let (X, d, μ) be a metric measure space. For ∅ = R ⊆ (0, ∞) consider the Hardy-Littlewood maximal operatorWe 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 and r > 0, then the weak (1, 1) norm of M R has the following localization property:An immediate consequence is that if (X, d, μ) is Ahlfors-David n-regular then the weak (1, 1) norm of M R 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 M R is proved by assigning to each f ∈ L 1 (X) a distribution over random martingales for which the associated (random) Doob maximal inequality controls the weak (1, 1) inequality for M R .

Dimension free bounds for the Hardy–Littlewood maximal operator associated to convex sets

“…By a clever covering argument with less overlap than in Vitali's lemma, they proved that the weak type constant admits a bound of the form O(n log n), and by using the Calderón-Zygmund method of rotations, they obtained for the strong type property a constant which behaves as np/(p−1). Concerning the weak type constant, Naor and Tao [60] have established the same n log n behavior for the large class of n-strong micro-doubling metric measure spaces (see also [25]). Several powerful results about the strong type constant for maximal functions associated to convex sets, beyond the one of Stein-Strömberg, have been established between 1986 and 1990.…”

Probabilistic clustering of high dimensional norms

“…Lemma B.1 below was proved in [MN07, Lemma 3.1] in the special case when α = 1/2, (X, d X ) is a finite metric space and µ is the counting measure on X (and with the factor 4 in the exponent in the right hand side of inequality (2.34) below replaced by the worse constant 8). We take this opportunity to record a more streamlined proof of [MN07, Lemma 3.1], though it is nothing more than a restructuring of the ideas of [MN07] (a similar argument was used in a slightly different setting in [NT10]). …”

Behavior of weak type bounds for high dimensional maximal operators defined by certain radial measures

“…Thus, for all p ≥ 1 and all (0, ∞). If μ is the measure defined by (15), then for every u ∈ (0, 1) and every R > 0 we have If additionally f is bounded, then for every u ∈ (0, 1)…”

“…A very significant extension of the Stein and Strömberg's O(d log d) result, beyond the Euclidean setting, has recently been obtained by A. Naor and T. Tao, cf. [15].…”