Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 163 / 4

Let $\Theta (R)$ partition an $R^{-1}$-neighborhood of the parabola into $R^{-1/2}\times R^{-1}$ caps, and suppose ${\hat{f}}_{\theta }$ is supported in $\theta$. The decoupling inequality for the parabola states that for $2\le p<\infty$ and every $ϵ>0$,

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For each $\tau \in \Theta (r)$ putMultiplication by $w_{r^{1/3}}$ enlarges Fourier support by at most $r^{-1/3}$ because ${\hat{w}}_{r^{1/3}}$ is supported in $B_{r^{-1/3}}(0)$. The supports of the $F_{\tau }$ therefore have uniformly bounded overlap: separated $r^{-1/3}$-scale moment-curve intervals remain disjoint except for boundedly many neighbors.

The fourier-support almost orthogonality supplied by the Plancherel theorem now gives

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SetFreeze $x_3$. In the $(x_1,x_2)$ Fourier variables, the functions $F_{\tau }(\cdot ,\cdot ,x_3)$ are supported in caps of tangential length $r^{-1/3}$ and normal width $r^{-2/3}$ along the parabola. Apply part a with decoupling scale $R^{\prime }=r^{2/3}$ and critical exponent $p=6$. Since $(R^{\prime }{)}^{1/2-3/6}=1$, for each fixed $x_3$,Integrate in $x_3$. Minkowski's inequality in $L^3(d{x}_3)$ giveswhich is exactly the claimed inequality after absorbing a change in $ϵ$.

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