Proposition 15

— Let $p$ and $q$ be two conjugate exponents ($1 < p < +\infty$). Assume that $\beta$ is left-invariant. Let $f \in L^p(G, \beta)$, $g \in L^q(G, \check{\beta})$. Then $f$ and $g$ are convolvable. The formula (15) defines, for every $x \in G$, a product $f * g$ that belongs to $\mathscr{K}(G)$ and is such that

For, we have $\dot{g} \in L^q(G, \beta)$, therefore the function $s \mapsto g(s^{-1}x)f(s)$ is $\beta$-integrable for every $x \in G$. Moreover,

therefore $f$ and $g$ are convolvable (Prop. 9). One sees at the same time that (15) defines for every $x$ a product $f * g$ such that

For $f, g$ in $\mathscr{K}(G)$, we have $f * g \in \mathscr{K}(G)$ (Prop. 11); therefore, for $f \in L^p(G, \beta)$ and $g \in L^q(G, \dot{\beta})$, the product $f * g$ furnished by (15) is the uniform limit of functions in $\mathscr{K}(G)$, hence belongs to $\overline{\mathscr{K}(G)}$.