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arXiv · 1908.08487

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

Samuel Zbarsky
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Abstract

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

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Samuel Zbarsky. 2019-08-22. Lower bounds and fixed points for the centered Hardy--Littlewood maximal operator. https://arxiv.org/abs/1908.08487

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