Periodic Fourier embeddings and products

The number of lattice points in the shell $n\le \mathrm{\mid }k\mathrm{\mid }<n+1$ is at most $C_d(n+1{)}^{d-1}$, by covering each point with a disjoint unit cube and bounding the volume of a shell of fixed enlarged width. Hence ${\sum }_k{w}_k^{-2\alpha }<\mathrm{\infty }$ whenever $\alpha >d\mathrm{/}2$. Cauchy–Schwarz gives ${\sum }_k\mathrm{\mid }k{\mathrm{\mid }}^j\mathrm{\mid }\widehat{q}(k)\mathrm{\mid }\le ({\sum }_k\mathrm{\mid }k{\mathrm{\mid }}^{2j}{w}_k^{-2\sigma }{)}^{1\mathrm{/}2}\mathrm{\parallel }q{\mathrm{\parallel }}_{H^{\sigma }}\le C\mathrm{\parallel }q{\mathrm{\parallel }}_{H^{\sigma }}$ when $\sigma >j+d\mathrm{/}2$.

The derivative Fourier series through order $j$ are absolutely and uniformly convergent by step 1.1. Their partial sums are derivatives of finite Fourier sums; the one-dimensional fundamental theorem of calculus in each coordinate identifies the uniform limits successively as the classical derivatives of the limit. This proves the $C^j$ representative and bound.

For polynomials, $\widehat{fg}(k)={\sum }_{\mathrm{\ell }}\widehat{f}(k-\mathrm{\ell })\widehat{g}(\mathrm{\ell })$. The weight inequality $w_{k+\mathrm{\ell }}^{\sigma }\le C_{\sigma }(w_k^{\sigma }+w_{\mathrm{\ell }}^{\sigma })$ follows from $w_{k+\mathrm{\ell }}\le w_k+w_{\mathrm{\ell }}$ and $(a+b{)}^{\sigma }\le C_{\sigma }(a^{\sigma }+b^{\sigma })$. Discrete Young, obtained by Minkowski's inequality on finite translates, yields $\mathrm{\parallel }fg{\mathrm{\parallel }}_{H^{\sigma }}\le C(\mathrm{\parallel }f{\mathrm{\parallel }}_{H^{\sigma }}\mathrm{\parallel }\widehat{g}{\mathrm{\parallel }}_{{\mathrm{\ell }}^1}+\mathrm{\parallel }\widehat{f}{\mathrm{\parallel }}_{{\mathrm{\ell }}^1}\mathrm{\parallel }g{\mathrm{\parallel }}_{H^{\sigma }})\le C\mathrm{\parallel }f{\mathrm{\parallel }}_{H^{\sigma }}\mathrm{\parallel }g{\mathrm{\parallel }}_{H^{\sigma }}$, since step 1.1 with $j=0$ bounds both coefficient ${\mathrm{\ell }}^1$ norms.

Finite Fourier sums are dense by the completion in [F1], so the polynomial product estimate extends uniquely by Cauchy convergence to $H^{\sigma }$. Since step 1.1 also gives convergence in $C^0$, this extended product agrees pointwise with multiplication of the continuous representatives. Applying the scalar estimate to finitely many components proves the vector and tensor versions.