Past exam of the mathematics course of the University of Cambridge / 2002 / iii / Paper 8 / 2 / i

Table of contents
- Solution i

Solution

The layer cake representation, followed by the assumed tail bound and Tonelli theorem, gives

\int_{X} f_{1}^{p} d μ = p \int_{0}^{\infty} a^{p - 1} μ {f_{1} > a} d a \leq p \int_{0}^{\infty} a^{p - 2} \int_{{g > a}} g d μ d a = p \int_{X} g (x) \int_{0}^{g (x)} a^{p - 2} d a d μ (x) = \frac{p}{p - 1} \int_{X} g^{p} d μ .

All integrands are nonnegative, so this use of Tonelli theorem is valid before finiteness of the left-hand side is known. Since

q = p / (p - 1)

, the result is

∥ f_{1} ∥_{p}^{p} \leq q ∥ g ∥_{p}^{p} .

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