I met an inequality in the book "Harmonic analysis and approximation on the unit sphere" by Wang Kunyang and Li Luoqing. I need some hints to follow the proof.
The proof is following,
\begin{align} \|T(f)\|_{q}&=\left\|\int_{\Omega}f(\eta)h(\xi\cdot\eta)d\eta\right\|_{q} \\ &\le\int_{\Omega}|f(\eta)|\left(\int_{\Omega}|h(\xi\cdot\eta)|^{q}d\xi\right)^{\frac{1}{q}}d\eta \end{align}
where $\Omega$ is the unit sphere, $f$ is a function defined on $\Omega$ and $h$ is a function defined on $[0,1]$. I think the domain does not affect the proof. I don't know how Minkowski's inequality applies here. As far as I know, Minkowski's inequality says this
$$\|f+g\|_{p}\le\|f\|_{p}+\|g\|_{p}$$
How could this applies in the proof?
I assume they are referring to the Minkowski's integral inequality: $$ \left\| \int_X F(x,y) \,d\mu(x)\right\|_q \leq \int_X \|F(x,y)\|_q \,d\mu(x). $$
where by $\| g \|_q$ I mean $\left(\int_Y |g(y)|^q \,d\nu(y)\right)^\frac{1}{q}$.