Young's Inequality for Convolutions/Corollary 2

Corollary to Young's Inequality for Convolutions

Let $f, g: \R^n \to \R$ be Lebesgue integrable functions.

Then their convolution $f * g$ is also Lebesgue integrable, and:

$\norm {f * g} \le \norm f \norm g$

Thus, convolution may be seen as a binary operation:

$*: \LL^1 \times \LL^1 \to \LL^1$

on the space of integrable functions $\LL^1$.

Proof

Apply Young's Inequality for Convolutions with $p = q = r = 1$.

$\blacksquare$