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

Samuel Zbarsky

paper

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

arXiv:1908.08487

Abstract

For all $p > 1$ and all centrally symmetric convex bodies $K \subset R^{d}$ define $M f$ as the centered maximal function associated to $K$ . We show that when $d = 1$ or $d = 2$ , we have $∣∣ M f ∣ ∣_{p} \geq (1 + ε (p, K)) ∣∣ f ∣ ∣_{p}$ . For $d \geq 3$ , 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 $q_{0} (K) > q_{0} (B (0, 1))$ .