Equality Case of Holder Inequality

Question

I was reading this post from 2011 on here regarding the equality case for the Holder inequality, i.e. $$\int |fg| =\|f\|_p\|g\|_q$$ First, the author of the post handled the cases where one of $\|f\|_p,\|g\|_q$ are either $0$ or $\infty$. From there, the author said

When we are proving the Hölder's inequality, we use that for $a,b\geq 0$ $$ab\leq \frac{a^p}{p}+\frac{b^q}{q}, \tag{1}$$ where the equality holds if and only if $b=a^{p/q}$. Explicitly $$\int\vert fg \vert\leq \Vert f \Vert_p \Vert g \Vert_q \int\left( \frac{\vert f \vert^p}{p\Vert f \Vert_p^p} + \frac{\vert g \vert^q}{q\Vert g \Vert_q^q}\right)=\Vert f \Vert_p \Vert g \Vert_q. \tag{2}$$

I was looking for some justification of the $\leq$ relationship in $(2)$. It's not clear to me how the author obtained this. Clearly, from $(1)$, it's not hard to see that the author was insisting that

$$\int|fg|\leq \int \frac{|f|^p}{p} + \frac{|g|^q}{q} \tag{3}$$

But from here, I get lost. I'm trying to reverse engineer $(2)$. So, $(2)$ can also be written as

$$\left( \frac{\|f\|_p^{p-1} \|g\|_q}{p}\right) \|f\|_p^p + \left( \frac{\|f\|_p\|g\|_q^{q-1}}{q} \right)\|g\|_q^q \tag{4}$$

But I suspect this manipulation is only making matters worse. How is (2) justified?

Answer 1

Define $F=\frac{|f|}{\|f\|_p}$. Then, $|f|=\|f\|_p\cdot F$, and $\int F^p=\int\frac{|f|^p}{\|f\|_p^p}=\frac{\|f\|_p^p}{\|f\|_p^p}=1$. Do a similar thing for $g$. So, \begin{align} \int|fg|&=\|f\|_p\|g\|_q\int FG\\ &\leq \|f\|_p\|g\|_q\int\left(\frac{F^p}{p}+\frac{G^q}{q}\right)\\ &= \|f\|_p\|g\|_q\cdot\left(\frac{1}{p}+\frac{1}{q}\right)\\ &= \|f\|_p\|g\|_q. \end{align} So, now if in the middle you plug in the definitions of $F$ and $G$ you see that term is precisely $\int\left(\frac{|f|^p}{p\|f\|_p^p}+ \frac{|g|^q}{q\|g\|_q^q}\right)$.

So, the point is that your statement $(3)$ is true, but it’s not exactly the optimal way of applying Young’s inequality. You actually can prove Holder’s inequality from this naive application of Young’s inequality, provided you follow it up by exploiting the symmetry imbalance. See Terry Tao’s writeup about amplification and arbitrage.