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:

$\left\Vert{f * g}\right\Vert \le \left\Vert{f}\right\Vert \, \left\Vert{g}\right\Vert$

Thus, convolution may be seen as a binary operation:

$*: \mathcal L^1 \times \mathcal L^1 \to \mathcal L^1$

on the space of integrable functions $\mathcal L^1$.

Proof

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

$\blacksquare$