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 $\map {L^p} {\R^n}$, $\map {L^q} {\R^n}$, and $\map {L^r} {\R^n}$ be Lebesgue spaces with seminorms $\norm {\, \cdot \,}_p$, $\norm {\, \cdot \,}_q$, and $\norm {\, \cdot \,}_r$ respectively.

Let $f \in \map {L^p} {\R^n}$ and $g \in \map {L^q} {\R^n}$.

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


 * $\norm {f * g}_r \le \norm f_p \cdot \norm  g_q$

Proof
We begin by seeking to bound $\size {\map {\paren {f * g} } x}$:

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 $(1)$, $(2)$ and $(3)$ in turn:

Therefore we have

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