Young's Inequality for Convolutions

Theorem
Let $f: \R^n \to \R$ be a Lebesgue integrable function.

Let $p \in \R$, $p \ge 1$, and let $g: \R^n \to \R$ be a Lebesgue $p$-integrable function.

Then the convolution $f * g$ of $f$ and $g$ satisfies:


 * $\left\Vert{f * g}\right\Vert_p \le \left\Vert{f}\right\Vert_1 \cdot \left\Vert{g}\right\Vert_p$

and hence is also Lebesgue $p$-integrable.

Here $\left\Vert{\cdot}\right\Vert_p$ denotes the $p$-seminorm.