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's Inequality for Integrals applied to three functions.

Note that the relation of conjugate exponents in the Generalized Hölder's Inequality for Integrals 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: