Proposition 13

Tag 0315, permanent. Cite it as [Bourbaki, Tag 0315].

Théories spectrales, chapter IV, § 3. Distributions et distributions tempérées

Let $p\in [1,+\infty ]$ . The space $\mathscr{S}(\mathbf{R}^n)$ is contained in $\mathscr{L}^p(\mathbf{R}^n)$ and the canonical injection of $\mathscr{S}(\mathbf{R}^n)$ into $\mathscr{L}^p(\mathbf{R}^n)$ is continuous. The image of $\mathscr{S}(\mathbf{R}^n)$ in $L^p(\mathbf{R}^n)$ is dense if $p\not = +\infty$ .

The first assertion is immediate for $p= +\infty$ . Henceforth suppose that $p\in [1,+\infty [$ . Let $m$ be an integer such that $n+ 1< mp$ . For every $\varphi \in \mathscr{S}(\mathbf{R}^n)$ and $x\in \mathbf{R}^n$ , one has

\|x\|^{n+1}|\varphi (x)|^p\leqslant \widetilde{q}_{0,m}(\varphi )^p

hence $\varphi \in \mathscr{L}^p(\mathbf{R}^n)$ by Prop. 3 of IV, p. 199. Moreover, one obtains

N_p(\varphi )\leqslant a^{1/p}_n\widetilde{q}_{0,0}(\varphi ) +b_n\widetilde{q}_{0,m}(\varphi )

where

a_n=\int d\mu(x),b_n=\int 1_{n+1}d\mu(x)

_{\|x\|\leqslant 1}\|_{x\|\geqslant 1}\|x\|

hence the injection of $\mathscr{S}(\mathbf{R}^n)$ into $\mathscr{L}^p(\mathbf{R}^n)$ is continuous.

As the space $\mathscr{D}(\mathbf{R}^n)$ is contained in $\mathscr{S}(\mathbf{R}^n)$ , Proposition 4 of IV, p. 202 implies that $\mathscr{S}(\mathbf{R}^n)$ is dense in $L^p(\mathbf{R}^n)$ if $p\not = +\infty$ .

Bourbaki sets the proof directly after the statement, and the corpus does not separate them, so what is above is the statement and its proof together.

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Cites

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