Simple case of Young's inequality

Question

I have a question concerning Young's inequality stated as follows:

$||a∗b||_{ℓ_q}≤||a||_{\ell_1}||b||_{ℓ_q},~~~~ 1≤q≤∞$.

Here you can find something on $\ell_q\big(\mathbb{Z}\big)$: Young's inequality for discrete convolution

Edit: From discussion in comments it is true for $\ell_q\big(\mathbb{N}\big)$ from the fact that it's true for $\ell_q\big(\mathbb{Z}\big)$. Does anyone can give straightforward proof for space $\ell_q\big(\mathbb{N}\big)$ without using $\ell_q\big(\mathbb{Z}\big)$ ?

Answer 1

You have that if $a \in l_q\big(\Bbb N\big)$ then $a \in l_q\big(\Bbb Z\big)$ since a sequence of natural numbers is a sequence of integers.

You also have if $a$ and $b$ are two sequences of natural numbers then so is $ a*b$, since the convolution only involves multiplications and additions.

So the equality you wrote holds for sequences of natural numbers too, not just sequences of integers.