Math 2.0

Harmonic analysis on Euclidean spaces Definition

Hardy–Littlewood maximal function

Builds on Lebesgue measureIntegrable functionOpen ball

Definition

Let n≥1n \ge 1, let λ\lambda be Lebesgue measure on ℝn\R^n, and let f ⁣:ℝn→ℝf \colon \R^n \to \R be locally integrable. For x∈ℝnx \in \R^n and r>0r > 0, let B(x,r)B(x, r) be the open ball of centre xx and radius rr. The Hardy–Littlewood maximal function of ff is the function Mf ⁣:ℝn→[0,∞]Mf \colon \R^n \to [0, \infty] given by Mf(x)=sup⁡r>01λ(B(x,r))∫B(x,r)∣f(y)∣ dλ(y).Mf(x) = \sup_{r > 0} \frac{1}{\lambda(B(x, r))} \int_{B(x, r)} \abs{f(y)} \, d\lambda(y).

Remarks

Example: let n=1n = 1 and f=𝟏[0,1]f = \mathbf{1}_{[0, 1]}, and let x>1x > 1. For r≥xr \ge x the average is 1/(2r)≤1/(2x)1/(2r) \le 1/(2x). For x−1<r<xx - 1 < r < x the average is (r−x+1)/(2r)(r - x + 1)/(2r), which increases with rr and tends to 1/(2x)1/(2x). For r≤x−1r \le x - 1 the average is 00. So Mf(x)=1/(2x)Mf(x) = 1/(2x).

Non-example of integrability: this shows that MfMf need not be integrable even when ff is, since ∫1∞dx/(2x)=∞\int_1^\infty dx/(2x) = \infty. In fact MfMf is integrable on ℝn\R^n only when f=0f = 0 almost everywhere.

Variants use balls containing xx instead of balls centred at xx (the uncentred maximal function), or cubes instead of balls. They are comparable to MfMf up to constants depending on nn.

Reading path

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Borel sigma-algebra, concept, no evidence levelBorel sigma-algebraconceptEuclidean space, object, no evidence levelEuclidean spaceobjectIntegrable function, concept, no evidence levelIntegrable functionconceptOpen ball, concept, no evidence levelOpen ballconceptOuter measure, concept, no evidence levelOuter measureconceptLebesgue measure, object, no evidence levelLebesgue measureobjectHardy–Littlewood maximal function, definition, no evidence levelHardy–Littlewood maxi…definition
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  1. Borel sigma-algebra concept
  2. Euclidean space object
  3. Integrable function concept
  4. Open ball concept
  5. Outer measure concept
  6. Lebesgue measure object
  7. Hardy–Littlewood maximal function definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Hardy–Littlewood_maximal_function
About this page
Tag744826NNN9
URIm2://744826NNN9
Kinddefinition
WritingW0 stub
MSC42B25, 28A15
DefinesHardy–Littlewood maximal function
UsesLebesgue measure, Integrable function, Open ball
Changed2 November 2026

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