 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2005/iii/paper-5/3/i/solution

Codex  0 Created 2026-10-03 Updated 2026-10-14

A nonnegative sublinear operator satisfies

S (f + g) \leq S (f) + S (g)

and

S (c f) = ∣ c ∣ S (f)

almost everywhere. It has weak type (1,1) if there is a finite constant

C

, independent of the function and threshold, such that

λ {x : S (f) (x) > α} \leq \frac{C}{α} ∥ f ∥_{L^{1} (R)} for every f \in L^{1} (R), α > 0.

For a signed or complex output use

∣ S (f) ∣

in the level set. The bound controls the distribution of large values; it does not assert that

S (f)

is integrable.

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