Young's Inequality for Convolutions/Corollary 1

Corollary to Young's Inequality for Convolutions

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

Let $p \in \R_{\ge 1}$.

Let $g: \R^n \to R$ be a Lebesgue $p$-integrable function.

Let $\left\Vert{f}\right\Vert_p$ denote the $p$-seminorm of $f$.

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.

Proof

Use Young's Inequality for Convolutions by letting $p'$, $q'$, and $r'$ be defined by $q'=r'=p$ and $p'=1$.

$\blacksquare$