Some remarks on dimension-free estimates for the discrete Hardy—Littlewood maximal functions

Abstract

Dependencies of the optimal constants in strong and weak type bounds will be studied between maximal functions corresponding to the Hardy—Littlewood averaging operators over convex symmetric bodies acting on ℝd and ℤd. Firstly we show, in the full range of p ∈ [1, ∞], that these optimal constants in Lp (ℝd) are always not larger than their discrete analogues in ℓp(ℤd); and we also show that the equality holds for the cubes in the case of p = 1. This in particular implies that the best constant in the weak type (1, 1) inequality for the discrete Hardy—Littlewood maximal function associated with centered cubes in ℤd grows to infinity as d → ∞, and if d = 1 it is equal to the largest root of the quadratic equation 12C2 − 22C + 5 = 0. Secondly we prove dimension-free estimates for the ℓp(ℤd) norms, p ∈ (1, ∞], of the discrete Hardy—Littlewood maximal operators with the restricted range of scales t ≥ Cqd corresponding to q-balls, q ∈ [2, ∞). Finally, we extend the latter result on ℓ2(ℤd) for the maximal operators restricted to dyadic scales 2n ≥ Cqd1/q.