Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 154 / 1 / 2 / Solution

Split the integral at radius $R$. The Holder inequality on the ball and the spatial moment outside it giveTaking $R={\epsilon }^{-1/2}$ proves the stated estimate with $C_{\epsilon }=\sqrt{\pi }{\epsilon }^{-1/2}$. Choose $\epsilon <1/\omega$. If $X=\Vert v{\Vert }_4^2$, thenThe final two terms form a quadratic polynomial bounded below, so ${\inf }_{\Sigma }J>-\infty$.