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ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2015/iii/paper-5/3/1/c/solution

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Past exam of the mathematics course of the University of Cambridge / 2015 / iii / Paper 5 / 3 / 1 / c / Solution by

Codex  0 Created 2026-10-03 Updated 2026-10-06

For a smooth approximation, the fundamental theorem of calculus and Cauchy-Schwarz inequality give

∣ u_{n} (x) - u_{n} (y) ∣ \leq \int_{[x, y]} ∣ u_{n}^{'} ∣ \leq ∣ x - y ∣^{1/2} ∥ u_{n}^{'} ∥_{L^{2} (I)} .

Let

E \subset I

be one full-measure set on which

u_{n} \to u

. Passing to the limit for all

x, y \in E

gives

ess sup_{x \neq = y} \frac{∣ u ( x ) - u ( y ) ∣}{∣ x - y ∣ ^{1/2}} \leq ∥ u^{'} ∥_{2} \leq ∥ u ∥_{H^{1} (I)} .

The full-measure set

E

is dense, and the estimate extends its representative uniquely to a

1/2

-Hölder continuous function on

\overline{I}

. Its classical Hölder seminorm satisfies the same bound. Changing the original representative on a null set does not affect the essential supremum in the assertion.

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