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

For all p > 1 and all centrally symmetric convex bodies, K. Rd defined Mf as the centered maximal function associated to K. We show that when d = 1 or d = 2, we have ||Mf || p = (1 + ( 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)).