I would like to apply Young's Convolution Inequality to the following functions:
$f,g \in L^{1}(\mathbb{R}^{N})$, where $f: \mathbb{R}^{N} \rightarrow \mathbb{R}^{N}$, $ g: \mathbb{R}^{N} \rightarrow \mathbb{R}$. That is, $f(x)$ is an $N$-vector, whilst $g(x)$ is a scalar, for all $x \in \mathbb{R}^{N}$.
My question is whether or not we can still apply Young's Inequality in this case. Is the convolution still well defined? I think the following definition is intuitive:
$ (f \ast g)(x) := \int_{\mathbb{R}^{N}} f(x-y)g(y) \text{d}y = \int_{\mathbb{R}^{N}} [f_{1}(x-y)g(y), ... ,f_{N}(x-y)g(y) ] \text{d}y$
The desired inequality is of course:
$ ||f \ast g||_{L^1} \leq ||f||_{L^1}||g||_{L^1} $
Using this definition, I followed the proof of Young's Inequality found in Haim Brezis's Functional Analysis textbook, and the proof doesn't seem to change whatsoever.