Centered Hardy–Littlewood maximal operator on the real line: Lower bounds

Abstract

For 1 < p < ∞ and M the centered Hardy-Littlewood maximal operator on R, we consider whether there is some ε = ε(p) > 0 such that ||M f || p ≥ (1+ε)||f || p . We prove this for 1 < p < 2. For 2 ≤ p < ∞, we prove the inequality for indicator functions and for unimodal functions. Résumé Soient 1 < p < ∞ et M la fonction maximale de Hardy-Littlewood sur R. Nouś etudions l'existence d'un ε = ε(p) > 0 tel que ||M f || p ≥ (1 + ε)||f || p . Nous l'établissons pour 1 < p < 2. Pour 2 ≤ p < ∞, nous prouvons l'inégalité pour les fonctions indicatrices et les fonctions unimodales.

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

“…Similar positive results have been obtained for dyadic maximal functions [6], maximal functions defined over λ-dense family of sets, and almost centered maximal functions (see [1] for details). This is closely related to the question of whether nonzero fixed points of M exist in L p ; in fact if (2) is satisfied, then no fixed points will exist. Korry [3] proved that the centered maximal operator for K a ball has fixed points if and only if d ≥ 3 and p > d d−2 , but a lack of fixed points does not imply that (2) holds.…”

“…This is closely related to the question of whether nonzero fixed points of M exist in L p ; in fact if (2) is satisfied, then no fixed points will exist. Korry [3] proved that the centered maximal operator for K a ball has fixed points if and only if d ≥ 3 and p > d d−2 , but a lack of fixed points does not imply that (2) holds. On the other hand, by comparing [1]), one can easily conclude that (2) holds true whenever p is sufficiently close to 1.…”

“…In [2], the authors work in d = 1 and prove (2) for all p < 1.5, and for all p for some special classes of functions f . Here, we prove it for all p < ∞ when d = 1 or d = 2, and also address the question of fixed points in dimension d > 2, showing that for a generic shape K, q 0 (K) > d d−2 = q 0 (B(0, 1)).…”