Lemma 16.15.197E8949E72BB2

Proof. By the inversion formula of theorem 2.9.4 and Fubini's theorem, $\widehat{vw}=\widehat v*\widehat w$. With $\langle \xi \rangle =(1+|\xi |^2)^{1/2}$ we have $\langle \xi \rangle \leq \langle \eta \rangle +\langle \xi -\eta \rangle $, since $(1,\xi )=(1,\eta )+(0,\xi -\eta )$ in $\mathbb R^{1+d}$; hence $\langle \xi \rangle ^ s\leq 2^ s(\langle \eta \rangle ^ s+\langle \xi -\eta \rangle ^ s)$, and therefore

Young's inequality $\| F*G\| _2\leq \| F\| _2\| G\| _1$ follows from the Cauchy–Schwarz inequality $|F*G(x)|^2\leq \| G\| _1\int |F(x-y)|^2|G(y)|\, \mathrm dy$ by integration in $x$ and Fubini's theorem. It therefore remains to bound $\| \widehat w\| _1$ and $\| \widehat v\| _1$ by the norms $\| \cdot \| _{\langle s\rangle }$. This follows from the Cauchy–Schwarz inequality, $\| \widehat w\| _1\leq \| \langle \cdot \rangle ^{-s}\| _2\| \langle \cdot \rangle ^ s\widehat w\| _2$, since $\langle \xi \rangle ^{-2s}$ is integrable on $\mathbb R^ d$ for $2s>d$. $\square$