How to prove$|\cdot|^{s}g\in\mathcal{S}'$? a question from Bahouri-Chemin-Danchin book "fourier analysis and nonlinear pde"

Question

In page 27 and 28 of book "fourier analysis and nonlinear partial differential equations", proposition 1.36 the authors Bahouri-Chemin-Danchin give a proof of this proposition, but I really don't know how to prove $|\cdot|^{s}g\in\mathcal{S}'(\mathbb{R}^{d})$? Any suggestions or hints are wellcome, thanks!

Answer 1

Recall $ f \in \mathcal{S} ' $ provided $ \lvert \langle f, \phi \rangle \rvert \leq C\sum \rho _{\alpha,\beta}(\phi) $ where $ \rho _{\alpha,\beta} $ are the seminorms and the sum is over a finite indexing set. Then as $g \in L^2 $ so $ \lvert \xi \rvert^s g$ is a function so it induces a distribution via integration so we just check $\forall \phi \in \mathcal{S} : $ \begin{align*} \lVert \lvert \xi \rvert^s g \phi \rVert_{L^1} &\leq \lVert \phi \rVert_{L^\infty} \lVert \lvert \xi \rvert^s g \rVert_{L^1(B_1)} + \lVert \langle \xi \rangle^N \phi \rVert_{L^\infty} \lVert \lvert \xi \rvert^s \langle \xi \rangle^{-N} g \rVert_{L^1(B_1^c)} \\ &\leq \lVert \phi \rVert_{L^\infty} \lVert \lvert \xi \rvert^s \rVert_{L^2(B_1)} \lVert g \rVert_{L^2} + \lVert \langle \xi \rangle^{ N} \phi \rVert_{L^\infty} \lVert \lvert \xi \rvert^{s - N} \rVert_{L^2(B_1^c)} \lVert g \rVert_{L^2} \end{align*} where we choose $N$ large enough so $\lVert \lvert \xi \rvert^{s - N} \rVert_{L^2(B_1^c)}< \infty$. So the only thing we have to make sure is finite is the $\lVert \lvert \xi \rvert^s \rVert_{L^2(B_1)}$ term. But $$ \lVert \lvert \xi \rvert^s \rVert_{L^2(B_1)} = \lVert \lvert \xi \rvert^{2s} \rVert_{L^1(B_1)}^{\frac{1}{2}} =C \lVert r^{2s + d - 1} \rVert_{L^1(B_1)}^{\frac{1}{2}} < \infty \iff 2s + d - 1 > - 1 \iff 2s + d > 0 \iff s > - \frac{d}{2} \impliedby \lvert s \rvert < \frac{d}{2}. $$