We denote by $H^{-1}(\mathbb{R}^N)$ the dual space to $H_0^1(\mathbb{R^N})$. Let $$\langle f,v \rangle = \int_{\mathbb{R}^N} fv \hspace{1cm} (f,v \in L^2(\mathbb{R}^N)).$$ It is known that if $f \in H^{-1}(\mathbb{R}^N)$, then there exists $f_0,f_1,\dots,f_N$ in $L^2(\mathbb{R}^N)$ such that $$\langle f , v\rangle = \int_{\mathbb{R}^N}f_0v + \sum_{i=1}^N f_i \frac{\partial v}{\partial x_i} \hspace{1cm} (v \in H_0^1(\mathbb{R})).$$ Starting from this specific definition of $H^{-1}(\mathbb{R}^N)$, I want to prove that $$H^{-1}(\mathbb{R}^N) = \{ f \in \mathcal{S}'(\mathbb{R}^N) : (1+\vert y \vert^2)^{-1/2} \widehat{u} \in L^2(\mathbb{R}^N) \},$$ where $\mathcal{S}'(\mathbb{R}^N)$ denotes the space of temperate distributions (dual space of the Schwartz space of rapidly decreasing functions).
Is there any book/paper where I can find this result? Most of the time, the above characterization by Fourier transform is taken as a definition...
Thanks!