Generalized Hölder inequality

Hint: $$ 1 = \frac{r}{p}+\frac{r}{q} = \frac{1}{\frac{p}{r}}+\frac{1}{\frac{q}{r}} $$

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 $\frac{r}{p}+\frac{r}{q}=1$.

we can remark that : $$\frac 1{p'} + \frac 1{q'} = 1$$ where : $$p'=\frac pr ; q'=\frac qr$$ and proof that $f^r \in L^{p'}$ and $g^r \in L^{q'}$