Lemma 44.2.3.label Let $G$ be a locally compact group and $f \in C_{c}(G; \complex)$, $f$ is a left convolver.
Proof. By Young’s Inequality, for each $g \in L^{2}(G; \complex)$, $\norm{f * g}_{L^2(G; \complex)}\le \norm{f}_{L^1(G; \complex)}\norm{g}_{L^2(G; \complex)}$.$\square$
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