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The centered and uncentered Hardy-Littlewood maximal functions

Definition

Assume the Axiom of Countable Choice (The Axiom of Countable Choice ( ${A C}_{ω}$ )).

Let $f \in L_{l o c}^{1} (R^{n})$ . The centered Hardy-Littlewood maximal function of $f$ is $M f (x) : = \sup_{r > 0} A_{r} ∣ f ∣ (x) = \sup_{r > 0} \frac{1}{λ (B (x, r))} \int_{B (x, r)} ∣ f (y) ∣ d λ (y) .$

The uncentered Hardy-Littlewood maximal function of $f$ is $M^{*} f (x) : = \sup_{x \in B (y, r)} \frac{1}{λ (B (y, r))} \int_{B (y, r)} ∣ f (z) ∣ d λ (z),$ where the supremum is over all Euclidean balls containing $x$ .

Both functions take values in $[0, \infty]$ .

Depends on

Used by

The Hardy-Littlewood maximal operator is not strong type (1,1) Counterexample
The maximal function of 1_[0,1] is not integrable Counterexample
The centered maximal function of 1_[0,1] on ℝ Example
The centered and uncentered maximal functions are pointwise comparable Proposition
The centered maximal operator is bounded on L^∞ Proposition
The centered Hardy-Littlewood maximal function is Borel measurable Theorem
The centered Hardy-Littlewood maximal operator is weak type (1,1) Theorem

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Sources

Gerald B. Folland, Real Analysis: Modern Techniques and Their Applications, 2nd ed., Theorem 3.17 and Exercise 22 (standard reference, not scraped)
Terence Tao, An Introduction to Measure Theory, Theorem 1.6.20 (standard reference, not scraped)