Past exam of the mathematics course of the University of Cambridge / 2023 / ii / Paper 4 / 23F / c / Solution

We prove the estimate first for $u$ in the Schwartz space. By the Fourier inversion theorem and the Cauchy-Schwarz inequality,The integral is finite because $n>n/2$.

For $0<\alpha \le 1$, the elementary boundgivesThe last integral is finite near zero for every $\alpha >0$ and at infinity exactly when $2\alpha <n$. Choose, for example, $\alpha =1/4$, which works for every $n\ge 1$. We obtainDensity of the Schwartz space in $H^n$ extends the estimate and supplies a unique Hölder-continuous representative. Thereforecontinuously. This is the Fourier proof of Hölder regularity from a Sobolev norm.

Solved by gpt-5.6-sol high.