If $p$ and $q$ are nonnegative numbers such that
$\frac{1}{p}+\frac{1}{q}=1$ and if $f \in L^p$ and $g \in L^q$, then $f\cdot g \in L^1$ and
$$\int |fg| \leqslant ||f||_p \cdot ||g||_q$$
I think Hölder's inequality is derived in order to prove Minkowski's inequality, which is a generalization of triangle inequality to $L^p$ norm. But is there any intuitive understanding of Hölder's inequality? It's hard for me to remember it. It seems that it's a generalization of the Cauchy-Schwarz inequality, trying to compare $L^2$ inner product to norm, but the power of each term is different, which makes it harder to be understood compared with Minkowski inequality.
Best Answer
Hölder's inequality is an attempt to generalize the Cauchy-Schwarz inequality to other Lebesgue exponents (other $L^p$ norm). In fact, for inequality of the form $$ \|fg\|_1 \leq \|f\|_p \|g\|_q$$ to be true, we must have $1/p+1/q=1$. To see this, we use scaling argument, which is also useful to verify the correct exponent for other type of inequalities (Sobolev inequalities, etc.).
If such inequality were true, then we could apply the inequality to functions $f(\lambda x)$ and $g(\lambda x)$ for $\lambda\in \mathbb{R}$ to get $$ \|fg\|_1 \leq \lambda^{n(1-1/p-1/q)} \|f\|_p \|g\|_q$$ which cannot be true for all $\lambda$ unless $1/p+1/q=1$. This is a way to see that indeed the correct exponents must be such that $1/p+1/q=1$.