Fourier transform wrt spatial variable of a distribution

Question

Let $T\in\mathcal{S}'(\mathbb{R}^{n+1})$ be a tempered distribution, i think that: $$ \mathbb{R}^{n+1}=\{(x,y):x\in\mathbb{R}^n,y\in\mathbb{R}\}.$$ How i can define the Fourier transform of $T$ wrt $x$? There is way to make this definition rigours? Any help is appreciated:

Answer 1

You can define the Fourier transform for only one variable as following:

If $T$ is a tempered distribution, $\mathbb{F}_yT$ is the only distribution such that for any $\psi \in \mathcal{S}(\mathbb{R}^{n+1})$ we have:

$$\langle\mathbb{F}_yT,\psi\rangle = \langle T,\mathbb{F}_y\psi\rangle$$

Where on the right hand side $\mathbb{F}_y\psi$ is the usual fourier tranform in only $y$ of $\psi$.