If $\frac{1}{p}+\frac{1}{q}=\frac{1}{r}$ then $\|fg\|_{L^r}\leq \|f\|_{L^p}\|g\|_{L^q}$

Question

If $\frac{1}{p}+\frac{1}{q}=\frac{1}{r}$ prove that $$\|fg\|_{L^r}\leq \|f\|_{L^p}\|g\|_{L^q}.$$ I tried to use Holder, but I can't prove it... any idea ?

Answer 1

I suppose that $fg\in L^r$ (and thus $|fg|^r\in L^1$), $f\in L^p$ and $g\in L^q$. Since $\frac{1}{p/r}+\frac{1}{q/r}=1$, using Holder you get $$\int |fg|^r\leq \left(\int |f|^{r\frac{p}{r}}\right)^{r/p}\left(\int |g|^{r\frac{q}{r}}\right)^{r/q}.$$

I let you finish.