I have a function $V\in W^{1,1}_\text{loc}(\mathbb{R}^n\times\mathbb{R}^+)$, such that, $V(x,\cdot)\in C^0([0,\infty))$, and: $V(x,0)=u(x)$, $\forall x\in\mathbb{R}^n$,where $u\in\mathcal{S}(\mathbb{R}^n)$ is given, moreover form some $a\in(-1,1)$: $$ \int_{\mathbb{R}^n\times\mathbb{R}^+}y^a|\nabla V(x,y)|^2\,dx\,dy<\infty, \tag{1}$$ by (1), we have that, for a.e. $y>0$, $\partial_{x_j}V(\cdot,y),\partial_y V(\cdot,y)\in L^2(\mathbb{R}^n),$ so make sense the Fourier transform $\mathcal{F}(\partial_yV(\cdot,y))\in L^2(\mathbb{R}^n)$, for a.e. $y>0$. What sense can I give to $\mathcal{F}(V(\cdot,y))$? There is way to prove that $V(\cdot,y)$ is a tempered distribution? In this case, how i can prove that: $$ \mathcal{F}(\partial_yV(\cdot,y))=\partial_y\mathcal{F}(V(\cdot,y))\tag{2}?$$ Morover, set: $\phi(\xi,y)=\mathcal{F}(V(\cdot,y))(\xi)$, $\xi\in\mathbb{R}^n,y>0$, fix $\xi\in\mathbb{R}^n$ and set $h(y)=\phi(\xi,|\xi|^{-1}y)$, is true that $h$ is continuos in $[0,\infty)$? How i can show that: $$ h'(y)=|\xi|^{-1}\partial_y\phi (\xi,|\xi|^{-1}y),$$ and deduce that: $h\in W^{1,1}_\text{loc}(0,\infty)$? I don't even know where to start. Please, help me.
These questions arise from the proof of the lemma 4.1.9 of "Some nonlocal operators and effects due to nonlocality", by C.Bocur, there is link.