Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 107 / 4 / b / i

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- Solution i

Solution

For

v = lo g u_{ε}

, Cauchy-Schwarz inequality and part (a.ii) imply

ρ^{1 - n} \int_{B_{ρ} (z) \cap B_{1}} ∣ D v ∣ \leq C (ρ^{2 - n} \int ∣ D v ∣^{2})^{1/2} \leq M (n, μ) .

The John-Nirenberg inequality therefore supplies

p_{0} = p_{0} (n, μ) > 0

such that, with

v_{B_{1}}

denoting the average,

\int_{B_{1}} e^{p_{0} ∣ v - v_{B_{1}} ∣} \leq C (n) .

Since

e^{p_{0} (v - v_{B_{1}})}

and

e^{- p_{0} (v - v_{B_{1}})}

are each bounded by this integrand,

(\int_{B_{1}} u_{ε}^{p_{0}}) (\int_{B_{1}} u_{ε}^{- p_{0}}) = (\int e^{p_{0} (v - v_{B_{1}})}) (\int e^{- p_{0} (v - v_{B_{1}})}) \leq C .

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