Hardy–Littlewood maximal function

The Hardy–Littlewood maximal function assigns to each point the supremum of a function’s averages over intervals or balls containing that point. It measures the function’s largest local concentration across scales.

Centered maximal function: The supremum of averages over intervals or balls centered at a given point. It restricts the averaging sets to be centered, unlike the uncentered version.
Lebesgue measure: A measure on Euclidean space that extends length, area, and volume to broad classes of sets. It measures both the averaging regions and the maximal function’s level sets.
Hardy–Littlewood maximal inequality: A measure bound for the set where the maximal function exceeds a positive threshold, controlled by the L1 norm. It is the basic estimate used to discard regions of excessive local average.
Strong maximal function: An operator taking suprema of averages over axis-parallel rectangles with independently varying side lengths. Rectangles allow more anisotropic averaging than the balls or intervals used here.
Uncentered maximal function: The supremum of averages over intervals or balls containing a point, with no requirement that they be centered there. This is the version defined by the stated containment rule.
Lebesgue integral: An integral defined using measure and approximation by measurable functions. The operator averages functions through integrals over measurable neighborhoods.
Calderón–Zygmund decomposition: A decomposition of an integrable function into good and bad parts using cubes where local averages exceed a threshold. Maximal-function level sets select the cubes in this decomposition.
Fractional maximal operator: An operator that weights local integrals by a power of the neighborhood’s size. Its scaling differs and it maps between different Lp spaces.
Vitali covering lemma: A covering principle that extracts a disjoint subcollection whose fixed enlargements cover the original family. It turns many high-average neighborhoods into a controlled measure estimate.
Lp space: A function space defined by finiteness of the p-th power integral, with the usual modification for p equal to infinity. Its norms provide the natural scale for maximal-operator bounds.

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