In the proof below, I'm confused how the authors got from the first to the second line and from the second to the third line. (These are indicated in the Red Box!)
I'm sure these are "easy" observations -- I'm just not seeing them.
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To go from the first to the second line, I assume the argument is to see the quantity as the $L^{q/p}(X_2,\mu_2)$ norm of the function $x_2 \mapsto \int_{X_1} f^p(x_1,x_2) \mathrm{d} \mu_1$. Then, they seem to use the classical general observation that $$ \| \varphi \|_{L^k} = \sup \left\{ \int \varphi \psi; \quad \psi \in L^{k'}, \|\psi\|_{L^{k'}} = 1 \right\}.$$ A similar argument is used once more to go from the third line to the fourth one.
To go from the second to the third line, the intuition is that the "best average choice" is smaller than the "average of the best choice". More formally, given any $g \in L^r$ and $x_1 \in X_1$, $$ \int_{x_2} f(x_1,x_2) g(x_2) \leq \sup_{\bar{g}} \int_{x_2} f(x_1,x_2) \bar{g}(x_2) $$ so you obtain the desired estimate by integrating over $x_1 \in X_1$ and taking the supremum over $g$ on the LHS.
