Local Lower Norm Estimates for Dyadic Maximal Operators and Related Bellman Functions

Abstract

We provide lower L q and weak L q -bounds for the localized dyadic maximal operator on R n , when the local L 1 and the local L p norm of the function are given. We actually do that in the more general context of homogeneous tree-like families in probability spaces.

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.…”

“…We now turn to understanding for what p we can prove (6). Suppose that we have a fixed point g ∈ L p (R d ) for d ≥ 3.…”

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…”