Lower bounds and fixed points for the centered Hardy--Littlewood maximal operator

2019/08/22 by Zbarsky, Samuel
#Classical Analysis and ODEs (math.CA) #FOS: Mathematics

paper · doi:10.48550/arxiv.1908.08487

Abstract

For all p>1 and all centrally symmetric convex bodies K⊂ ℝ^d define Mf as the centered maximal function associated to K. We show that when d=1 or d=2, we have ||Mf||_p≥ (1+ε(p,K))||f||_p. For d≥ 3, let q₀(K) be the infimum value of p for which M has a fixed point. We show that for generic shapes K, we have q₀(K)>q₀(B(0,1)).

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