= Solution
Tonelli's theorem and the triangle inequality give
$$
\begin{aligned}
\int_0^\infty |G(y)g(y)|\,dy
&\leq\int_0^\infty\int_0^\infty
|F(x,y)g(y)|\,dx\,dy\\
&=\int_0^\infty\left(\int_0^\infty
|F(x,y)g(y)|\,dy\right)dx.
\end{aligned}
$$
Applying <Holder inequality> in the $y$ variable for each fixed $x$ yields
$$
\int_0^\infty |G(y)g(y)|\,dy
\leq
\int_0^\infty
\left(\int_0^\infty |F(x,y)|^p\,dy\right)^{1/p}
\|g\|_q\,dx,
$$
and $\|g\|_q\leq1$ gives the claimed inequality. This is the duality proof of the <Minkowski integral inequality>.
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