I got stuck with the following problem: prove that if $p,q \geqslant 1$ with $pq/(p+q)\geqslant 1$, $u \in L^p$ and $v \in L^q$, then $uv \in L^{pq/(q+p)}$ and $\left \| uv \right \|_{L^{pq/(q+p)}} \leqslant \left \| u \right \|_{L^p}\left \| v \right \|_{L^q}$.
My idea is to make computations on $\int |uv|$ but the problem is that $\frac{1}{p}+\frac{1}{q}\neq 1$, so I guess I can't apply Holder and/or Minkowski inequalities.
Hints would be preferable than complete answers..
Denote by $\lambda$ the quotient $(p+q)/pq$. Observe that $\lambda p$, $\lambda q>1$ and $\frac 1{\lambda p}+\frac 1{\lambda q}=1$. Hölder's inequality yields $$\|u\|_p\|v\|_q=\left(\Big\||u|^{1/\lambda}\Big\|_{\lambda p}\Big\||v|^{1/\lambda}\Big\|_{\lambda q}\right)^\lambda\ge\Big\||uv|^{1/\lambda}\Big\|^\lambda=\|uv\|_{1/\lambda}$$