There is a Young inequality that says: given $a,b \in \mathbb{R}$ and $p,q> 1$ such that $\frac{1}{p} + \frac{1}{q}=1$ we have:

$$|ab| \leq \frac{1}{p}|a|^p + \frac{1}{q} |b|^q$$

The question is: given two real functions $f,g: [a,b] \to \mathbb{R}$ (let's take them continuous to avoid integrability problems) the inequality above should hold for every $x$ and thus pass to the integral (at least I don't see why it shouldn't). So I should have something like:

$$\int_{a}^{b} |f(x)g(x)| dx \leq \frac{1}{p} ||f||_{L^p} ^p + \frac{1}{q} ||g||_{L^q}^q$$

But I have never seen this, in this case I often use Holder for instance. Is this inequality true?


It is true. The reason why you see just Holder's used more often, is because your inequality is equivalent to applying Holder's and Young's in succession. Indeed: $$\int_a^b |f(x)g(x)|dx\leq \|f\|_{L^p}\|g\|_{L^q}\leq\frac{1}{p}\|f\|_{L^p}^p+\frac{1}{q}\|g\|_{L^q}^q $$ The first inequality is Holder's while the second is Young's with $a=\|f\|_{L^p}$ and $b=\|g\|_{L^q}$.

Most of the times you just need Holder's inequality so you stop at the first step.

Also, Holder's inequality works (and is very often used) for the case $p=1,q=\infty$ (or viceversa) while this one does not.


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