Past exam of the mathematics course of the University of Cambridge / 2009 / iii / Paper 9 / 2 / b

Solution

If either upper norm bound is nonpositive, there are no admissible functions with

∥ f ∥_{q} \geq C_{q} > 0

, and the assertion is vacuous. For positive

C_{p}, C_{r}

, choose

ε = (\frac{C _{q}^{q}}{2 C _{p}^{p}})^{1/ (q - p)}, A = {x : ∣ f (x) ∣ > ε} .

Splitting the integral gives

C_{q}^{q} \leq ∥ f ∥_{q}^{q} \leq ε^{q - p} ∥ f ∥_{p}^{p} + \int_{A} ∣ f ∣^{q} \leq \frac{C _{q}^{q}}{2} + \int_{A} ∣ f ∣^{q} .

For finite

r

, Holder inequality bounds the last term by

∥ f ∥_{r}^{q} μ (A)^{1 - q / r} \leq C_{r}^{q} μ (A)^{1 - q / r}

. Hence

μ (A) \geq (\frac{C _{q}^{q}}{2 C _{r}^{q}})^{r / (r - q)} .

For

r = \infty

, the corresponding bound is

\int_{A} ∣ f ∣^{q} \leq C_{r}^{q} μ (A)

. Thus an explicit choice proving the strict inequality is

M = {\frac{1}{2} (C_{q}^{q} / (2 C_{r}^{q}))^{r / (r - q)}, C_{q}^{q} / (4 C_{r}^{q}), r < \infty, r = \infty. μ {∣ f ∣ > ε} > M .

p

bound prevents diffuse low-amplitude mass, and the higher-

r

bound prevents arbitrarily concentrated spikes.

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