I was reading the answer to this question here.
I am having trouble proving $\|fgh\|\leq\|f\|_p\|g\|_q\|h\|_r$, where $\frac{1}{p}+\frac{1}{q}+\frac{1}{r}=2$ and I assume $f\in L_p,$ $g\in L_q$, $h\in L_r$.
The rest of the argument is very clean I'm just stuck on this part. I know the proof for $\frac{1}{p}+\frac{1}{q}+\frac{1}{r}=1$, but I am at a loss for how to start when the $1$ is replaced by $2$.
Any help on how to get going would be appreciated.
The answer in that link has now been fixed to be correct (and the inequality is different to the one in the question), but I'll leave this here as the scaling argument is quite interesting.
The only norm that can go on the LHS with that relation is a $1/2$-seminorm, by a scaling argument (replace all functions by the same function evaluated at $\lambda x$). If that inequality were true, then for functions $f,g,h$ which are non zero and $fgh$ is non zero, applying the inequality to $f(\lambda x), g(\lambda x),h(\lambda x)$ for $\lambda \in \mathbb{R}$, you get $$\lambda^{-n}\|fgh\|_1 \leq \lambda^{-n(1/p + 1/q + 1/r)}\|f\|_p\|g\|_q\|h\|_r,$$ which can't be true for every $\lambda$ unless $1/p + 1/q + 1/r = 1$.
The way I would prove Young's inequality is to apply Riesz-Thorin to the operator $g \mapsto f* g$ for fixed $f$.
