Young's Inequality for Convolutions

Theorem
Let $p, q, r \in \R_{\ge 1}$ satisfy:
 * $1 + \dfrac 1 r = \dfrac 1 p + \dfrac 1 q$

Let $L^p \left({\R^n}\right)$, $L^q \left({\R^n}\right)$, and $L^r \left({\R^n}\right)$ be Lebesgue spaces with seminorms $\left\Vert{\cdot}\right\Vert_p$, $\left\Vert{\cdot}\right\Vert_q$, and $\left\Vert{\cdot}\right\Vert_r$ respectively.

Let $f \in L^p \left({\R^n}\right)$ and $g \in L^q \left({\R^n}\right)$.

Then the convolution $f * g$ is in $L^r \left({\R^n}\right)$ and the following inequality is satisfied:


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

Proof
We begin by seeking to bound $\left\vert{\left({f * g}\right) \left({x}\right)}\right\vert$:

where the last inequality is via the Generalized Hölder Inequality applied to three functions.

Note that the relation of conjugate exponents in the Generalized Hölder Inequality is satisfied:

We now analyze terms $I$, $II$, and $III$ individually:

With these preliminary calculations out of the way, we turn to the main proof: