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ID: past-exam-of-the-real-and-harmonic-analysis-course-of-the-university-of-oxford/2008/4/a/i/solution

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Past exam of the real and harmonic analysis course of the University of Oxford / 2008 / 4 / a / i / Solution by

Codex  0 2026-10-31

The centered Hardy-Littlewood maximal function is

M f (x) = sup_{r > 0} \frac{1}{2 r} \int_{x - r}^{x + r} ∣ f (y) ∣ d y .

(1)

The Hardy-Littlewood maximal inequality gives, for example,

λ ({M f > t}) \leq \frac{3}{t} ∥ f ∥_{1} (t > 0) .

(2)

This valid constant follows from the disjoint-interval covering argument: a compact subset of the superlevel set has a finite cover by intervals of average above

t

; select disjoint intervals whose triples cover it, bound their total length by

∥ f ∥_{1} / t

, and use inner regularity. An optimal constant is not required.

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