Proposition 30.4.8.label Let $G$ be a locally compact group, $p \in [1, \infty)$, and $E$ be a normed vector space over $K \in \RC$, then
- (1)
The mapping $G \times L^{p}(G; E) \to L^{p}(G; E)$ defined by $(x, f) \mapsto L_{x}f$ is jointly continuous.
- (2)
The mapping $G \times L^{p}(G; E) \to L^{p}(G; E)$ defined by $(x, f) \mapsto R_{x}f$ is jointly continuous.
Proof, [Proposition 2.42, Fol16]. (1): Let $\eps > 0$, $x, y \in G$, and $f, g \in L^{p}(\mu; E)$, then
By Proposition 25.1.7, there exists $\phi \in C_{c}(G; E)$ such that $\norm{\phi - f}_{L^p(\mu; E)}< \eps$. In which case,
By Proposition 30.1.2, there exists $V \in \cn_{G}(1)$ such that if $x^{-1}y \in V$, then $\norm{L_{x^{-1}y}\phi - \phi}_{u} < \eps/(2\mu\bracs{\phi \ne 0}^{1/p})$. Thus if $x^{-1}y \in V$, then
(2): Let $V \in \cn_{G}(1)$ be compact, then since $\Delta_{G}: G \to (0, \infty)$ is a continuous homomorphism, $C := \sup_{y \in V}\Delta_{G}(y^{-1}) < \infty$ by Proposition 5.16.3. For any $f \in L^{p}(\mu; E)$ and $x \in V$,
Let $f, g \in L^{p}(G; E)$ and $x \in V$, then
By Proposition 25.1.7, there exists $\phi \in C_{c}(G; E)$ with $\norm{f - \phi}_{L^p(G; E)}< \eps$. Thus
Since $V$ is compact, $\bracsn{\phi \ne 0}V^{-1}$ is relatively compact, and of finite measure. By Proposition 30.1.2, $\phi$ is left and right uniformly continuous, so there exists $W \in \cn_{G}(1)$ such that $W \subset V$ and $\norm{R_x\phi - \phi}_{u} < \eps/\normn{\one_{\bracsn{\phi \ne 0}V^{-1}}}_{L^p(G; \real)}$. In which case, for any $x \in W$,
and
$\square$
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