Past exam of the mathematics course of the University of Cambridge / 2009 / iii / Paper 8 / 3 / b

Write $⟨k⟩=(1+|k{|}^2{)}^{1/2}$. There are $O(2^{jn})$ lattice points in the dyadic shell $2^j\le |m|<2^{j+1}$, while each contributes at most a constant times $2^{-2js}$. The geometric series ${\sum }_j{2}^{j(n-2s)}$ converges when $2s>n$, proving ${\sum }_m⟨m{⟩}^{-2s}<\infty$. For $|\alpha |\le k$ and $u\in H_{s+k}$, the Cauchy-Schwarz inequality yieldsThus every derivative Fourier series through order $k$ converges absolutely and uniformly. Starting with the continuous sum for $u$, uniform convergence of successive derivative series justifies differentiation, so these are its continuous classical derivatives. $\boxed{H_{s+k}({\mathbb{T}}^n)\hookrightarrow C^k({\mathbb{T}}^n),s>n/2}$, with a continuous norm bound.