Exercise 13, § 4

Let $G$ be a locally compact group operating continuously on the left in a Polish locally compact space $T$. Let $\beta$ be a left Haar measure on $G$, $\nu$ a quasi-invariant positive measure on $T$, and $\chi(s, x)$ a function $> 0$ on $G \times T$ satisfying the conditions of Exer. 13 of Ch. VII, §1. For $f \in L^p(T, \nu)$ ($1 \leq p < +\infty$), set

a) Show that for every $s \in G$, $\gamma_{\chi, p}(s)$ is an isometric endomorphism of $L^p(T, \nu)$. Show that the mapping $s \mapsto \gamma_{\chi, p}(s)$ is a linear representation of $G$ in $L^p(T, \nu)$ (argue as in §2, No. 5). b) Let $f \in \mathscr{L}^p(T, \nu)$, and let $h \in \mathscr{K}(G)$. Show that the function is $p$-th power integrable for $\beta \otimes \nu$ (begin with the case $p = 1$, and use Lemma 1 of §4, No. 1). Show that if $q$ is the exponent conjugate to $p$, and if $g \in \mathscr{L}^q(T, \nu)$, then $f(s^{-1}x)h(s)g(x)\chi(s^{-1}, x)^{1/p}$ is integrable for $\beta \otimes \nu$ (write $h$ in the form $h_1h_2$ with $h_1, h_2$ in $\mathscr{K}(G)$) . Then deduce from the Lebesgue–Fubini theorem that, for $f \in L^p(T, \nu)$ and $g \in L^q(T, \nu)$, the function is $\beta$-measurable.

c) Show, using Cor. 2 of Prop. 18 of §4, No. 6 and Lemma 1 of Ch. VI, §3, No. 1, that the representation $s \mapsto \gamma_{\chi,p}(s)$ of G in $L^p(T, \nu)$ is continuous for $1 \leq p < +\infty$.

d) Assume in addition that, for every $s \in G$, the function $x \mapsto \chi(s^{-1}, x)$ is bounded. For $f \in L^p(T, \nu)$, set

Show that $s \mapsto \gamma_{\chi}(s)$ is a continuous representation of G in $L^p(T, \nu)$ for $1 \leq p < +\infty$. (One shows, as in the proof of Prop. 9 of §2, No. 5, that $s \mapsto \gamma_{\chi}(s)$ is a representation of G by endomorphisms of $L^p(T, \nu)$. One then observes that if $f \in L^p(T, \nu)$ and $h \in \mathscr{K}(G)$, Lemma 1 of §4, No. 1 shows that $h(s)f(s^{-1}x)\chi(s^{-1}, x)$ is $p$-th power integrable for $\beta \otimes \nu$. The proof is concluded as in c.)