Proposition 9

Suppose that each function $\chi(s, \cdot)$ is bounded. Then $\gamma_\chi$ leaves $L_C^p(X, \beta)$ stable, and the linear representation $\gamma_\chi$ of $G$ in $L_C^p(X, \beta)$ is continuous.

Let $f \in \mathscr{L}_C^p(X, \beta)$. Then

therefore $\gamma_\chi(s)f \in \mathscr{L}_C^p(X, \beta)$, and (5) where $q$ denotes the exponent conjugate to $p$. If $f \in \mathscr{K}(X)$, then $\gamma_{\chi}(s)f$ tends to $\gamma_{\chi}(s_0)f$ in $\mathscr{K}(X)$, hence in $\mathscr{L}_C^p(X, \beta)$, as $s$ tends to $s_0$. Therefore the representation $\gamma_{\chi}$ of $G$ in $L_C^p(X, \beta)$ is continuous (No. 1, Prop. 2).

Properties analogous to those of Nos. 3, 4, 5 hold if $G$ operates on the right in $X$.

In particular, if one regards $G$ as operating on itself by left or right translations, and if one takes $\chi = 1$, one obtains the left and right regular representations of $G$ in $\mathscr{C}(G)$, $\mathscr{K}(G)$, $\overline{\mathscr{K}}(G)$, $\mathscr{C}'(G)$, $\mathscr{M}(G)$, $\mathscr{M}^1(G)$. If one takes $\beta$ to be a left (resp. right) Haar measure on $G$, and if one takes $\chi = 1$, one obtains the left (resp. right) regular representation of $G$ in $L_C^p(G, \beta)$.