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 $\norm f_p$ denote the $p$-seminorm of $f$.

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


 * $\norm {f * g}_p \le \norm f_1 \cdot \norm g_p$

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 $p' = 1$.