TITLE: Prove that $fg\in L^r(\Omega)$ if $f\in L^p(\Omega),g\in L^q(\Omega)$, and $\frac1 p+\frac1 q=\frac1 r$
QUESTION [16 upvotes]: Can anyone give me a hint for proving the following:
Let $\Omega$ be a measure space. Assume $f \in L^p(\Omega)$ and $g \in L^q(\Omega)$ with $1 \leq p, q \leq \infty$ and $\frac1p + \frac1q \leq 1$. Prove that $fg \in L^r(\Omega)$ with $\frac1r = \frac1p + \frac1q$. 
Note: One should be able to use (the standard) Hölder inequality. Notice that if you have $\frac1p + \frac1q = 1$ you recover the former result.

REPLY [21 votes]: Using the standard Hölder inequality:
$$
\|fg\|_r^r=\|f^rg^r\|_1\le\|f^r\|_{p/r}\|g^r\|_{q/r}=\|f\|_p^r\|g\|_q^r
$$
Since $\frac{r}{p}+\frac{r}{q}=1$.