Lower Bounds for Uncentered Maximal Functions in Any Dimension

Abstract

Abstract. In this paper we address the following question: given p ∈ (1, ∞), n ≥ 1, does there exists a constantwhere M f is a maximal function operator defined over the family of shifts and dilates of a centrally symmetric convex body. The inequality fails in general for the centered maximal function operator, but nevertheless we give an affirmative answer to the question for the uncentered maximal function operator and the almost centered maximal function operator. In addition, we also present the Bellman function approach of Melas, Nikolidakis and Stavropoulos to maximal function operators defined over various types of families of sets, and in case of parallelepipeds we will show that

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.…”

“…This question was asked for the uncentered maximal operator with K a ball by Lerner in [4] and answered affirmatively for all p < ∞ in [1]. In fact they showed this for general centrally symmetric convex K and for uncentered maximal function defined by taking…”

“…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 M f (x) ≥ C(d)M u f (x), and using the fact that [1]), one can easily conclude that (2) holds true whenever p is sufficiently close to 1. It is natural to ask what is the maximal p 0 (K) for which if 1 < p < p 0 (K) then (2) holds.…”

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

“…The paper also studied the estimate (2) for other maximal functions. For example, the lower bound (2) persists if one takes supremum in (1) over the shifts and dilates of a fixed centrally symmetric convex body K. Similar positive results have been obtained for dyadic maximal functions [5]; maximal functions defined over λ-dense family of sets, and almost centered maximal functions (see [2] for details). The Lerner's inequality for the centered maximal function…”

“…In fact, Korry [3] proved that the centered maximal operator does not have fixed points unless n ≥ 3 and p > n n−2 , but a lack of fixed points does not imply that (3) holds. On the other hand for any n ≥ 1, by comparing M f (x) ≥ C(n)M u f (x), and using the fact that [2]), one can easily conclude that (3) holds true whenever p is sufficiently close to 1. It is natural to ask what is the maximal p 0 (n) for which if 1 < p < p 0 (n) then (3) holds.…”