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

Abstract

For all $p>1$ and all centrally symmetric convex bodies, $K\subset {\mathbb {R}}^d$ defined Mf as the centered maximal function associated to K. We show that when $d=1$ or $d=2$, we have $||Mf||_p\ge (1+\epsilon (p,K))||f||_p$. For $d\ge 3$, 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 $q_0(K)>q_0(B(0,1))$.

A Besicovitch Covering Lemma

Lemma 5

Let $K\subset {\mathbb {R}}^d$ be convex, compact, and centrally symmetric. Suppose we have a bounded set E and some constant $\Lambda >0$, and take

$$\begin{aligned} A=\{x+\lambda _x K\}_{x\in E} \end{aligned}$$

where $0<\lambda _x<\Lambda$ for each $x\in E$. Then there is some countable ${{\tilde{A}}}\subset A$ so that the sets ${{\tilde{A}}}$ cover all of E, and each point is covered at most B(d) times, where B(d) is a constant depending only on d.

Note that this formulation is the same as the formulation for balls in some norm on ${\mathbb {R}}^d$. The author has not found a place where it is clearly stated with a proof, but a proof can be obtained by a straightforward modification of the standard proof of the Bescovitch covering lemma for the usual norm on ${\mathbb {R}}^n$, which can be found for instance in [6]. The appendix of [3] also gives references which they claim can be found to contain the proof of the Besicovitch Covering Lemma for arbitrary norm.