Lower Bounds and Fixed Points for the Centered Hardy–Littlewood Maximal Operator

Abstract

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