In Grafakos' Classical Fourier Analysis (third edition) on p. 40 the following example of an application of the Riesz-Thorin interpolation theorem can be found:
Young's inequality convolution
The bound of $T: L^1 \rightarrow L^r$ is clear by Minkowskis integral inequality. However, I do not see, why $T:L^{r'} \rightarrow L^\infty$ is bounded by $\|g\|_{L^r}\|f\|_{L^{r'}}$. Since $r'$ is the conjugate exponent of $r$, $1\leqslant r \leqslant \infty$, Hölder's inequality may be used, but I do not see how. Grafakos defines the convolution on locally compact groups $G$ with left invariant Haar measure denoted by $\lambda$ such that $(G,\lambda)$ is $\sigma$-finite by
$\displaystyle (f \ast g)(x) := \int_G f(y)g(y^{-1}x)d\lambda(y)$
I think the best would be to show that $\vert f \ast g \vert$ is bounded.