Lemma 7

Tag 03CU, permanent. Cite it as [Bourbaki, Tag 03CU].

Théories spectrales, chapter V, § 3. Fonctions de type positif

Let $\psi \in L^1(G)$ be such that $\psi \geqslant 0$ and $\int\psi = 1$ . Let us denote by $p$ the seminorm on $\mathscr{C}_b(G)$ defined by $p(\varphi ) =|\langle \varphi , \psi \rangle |$ for all $\varphi \in \mathscr{C}_b(G)$ . For every $\varphi \in$ Pos $_1(G)$ , one has

\|\psi *\varphi -\varphi \|_{\infty}\leqslant \surd\overline{2p(1 -\varphi)}

Let $\varphi \in$ Pos $_1(G)$ and $x\in G$ . By INT, VIII, p. 167, § 4, n $^o5$ , prop. 14, we obtain

|\psi *\varphi (x)-\varphi (x)|=|\int_G(\varphi (y^{-1}x)-\varphi (x))\psi (y)d\mu(y)|

\leqslant \int_G|\varphi (y^{-1}x)-\varphi (x)|\psi (y)d\mu(y)

\leqslant \int_G\surd\overline{2(1 -\mathscr{R}\varphi(y))}\psi (y)d\mu(y)

applying the majorization (1) of V, p. 446. The Cauchy-Schwarz inequality then implies

\surd \int 1/2\int 1/2

\|\psi *\varphi -\varphi \|_{\infty}\leqslant \overline{2}(_G(1-\mathscr{R}(\varphi ))\psi d\mu)(_G\psi d\mu)

\surd \surd

$\leqslant$ 2 $p(1-\varphi )$ ,

whence the result.

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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