2020/10/14 by Kosz, Dariusz, Mirek, Mariusz, Plewa, Paweł +1 · 1 citation
#Classical Analysis and ODEs (math.CA) #FOS: Mathematics
paper · doi:10.48550/arxiv.2010.07379
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 \mathbb Rd and \mathbb Zd. Firstly, we show, in the full range of p∈[1,∞], that these optimal constants in Lp(\mathbb Rd) are always not larger than their discrete analogues in ℓp(\mathbb 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 \mathbb 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(\mathbb Zd) norms, p∈(1,∞], of the discrete Hardy--Littlewood maximal operators with the restricted range of scales t≥ Cq d corresponding to q-balls, q∈[2,∞). Finally, we extend the latter result on ℓ2(\mathbb Zd) for the maximal operators restricted to dyadic scales 2n≥ Cq d1/q.