Lemma ((474))

Assume $p>1$ and $q>1$ are s.t. $\frac{1}{p}+\frac{1}{q}=1$. Let $u(x)\ge 0$ and $v(x)\ge 0$ satisfy $$\int\limits _{-\infty}^{\infty}u(x)^{p} \, dx \ \ \ \int\limits _{-\infty}^{\infty}v(x)^{q} \, dx$$Then $$\int\limits _{-\infty}^{\infty}u(x)v(x) \, dx \le \left( \int\limits _{-\infty}^{\infty} u(x)^{p} \, dx \right)^{1/p}\left( \int\limits _{-\infty}^{\infty} v(x)^{q} \, dx \right)^{1/q}$$or $$\int\limits _{-\infty}^{\infty}u(x)v(x) \, dx \le \|u\|_{p}\|v\|_{q}$$Moreover, equality holds if and only if $v(x)^{q}=Cu(x)^{p}$ for some $C>0$.