Theorem 30.4.5 (Young’s Inequality).label Let $G$ be a locally compact group, $f \in L^{1}(G; \complex)$, $p \in [1, \infty]$, and $g \in L^{p}(G; \complex)$, then
- (1)
If $p = \infty$, then $(f * g)(x) = \int_{G} f(y)g(y^{-1}x)dy$ exists for all $x \in G$, with
\[\norm{f * g}_{L^\infty(G; \complex)}\le \norm{f}_{L^1(G; \complex)}\norm{g}_{L^\infty(G; \complex)}\] - (2)
If $p \in [1, \infty)$, then $(f * g)(x)$ exists for almost every $x \in G$, and $f * g \in L^{p}(G; \complex)$ with
\[\norm{f * g}_{L^p(G; \complex)}\le \norm{f}_{L^1(G; \complex)}\norm{g}_{L^p(G; \complex)}\]
Proof, [Proposition 2.40, Fol99]. (1): By Definition 30.3.1, the measures $dx$ and $dx^{-1}$ are equivalent. Thus for each $x \in G$, $\normn{y \mapsto g(y^{-1}x)}_{L^\infty(G; \complex)}= \norm{g}_{L^\infty(G; \complex)}$. By Hölder’s Inequality
(2): Since $\bracsn{f \ne 0}$ and $\bracsn{g \ne 0}$ are $\sigma$-finite, assume without loss of generality that $dx$ is $\sigma$-finite. In which case, by Minkowski’s Inequality for integrals,
$\square$
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