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

For $v=\log u_{\epsilon }$, Cauchy-Schwarz inequality and part (a.ii) implyThe John-Nirenberg inequality therefore supplies $p_0=p_0(n,\mu )>0$ such that, with $v_{B_1}$ denoting the average,Since $e^{p_0(v-v_{B_1})}$ and $e^{-p_0(v-v_{B_1})}$ are each bounded by this integrand,