Define the Fourier transform on $f \in L^1(\mathbb{R}^N;\mathbb{C})$ by $$\hat{f}(\xi)= \int_{\mathbb{R}^n} f(x) e^{- i \xi \cdot x} \, \mathrm{d}x.$$
Among several properties, we have:
(i) If $f \in L^1(\mathbb{R}^N;\mathbb{C})$ is such that $x_k f \in L^1(\mathbb{R}^N;\mathbb{C})$ ( $1\leq k \leq n$ ), then $\hat f$ is differentiable with respect to $\xi_k$ and $\partial_k \hat f = \widehat{-i x_k f}$.
(ii) Conversely, if $f \in L^1(\mathbb{R}^N;\mathbb{C})$ such that its weak derivative $\partial_k f \in L^1(\mathbb{R}^N;\mathbb{C})$, then $\widehat{\partial_k f}= i \xi_k \widehat{f}$.
Now, we extend the Fourier transform on $L^2$ using the classic $L^1 \cap L^2$ density approach. The above property should adapt to the this extension in the following way:
(iii) If $f \in L^2(\mathbb{R}^N;\mathbb{C})$ is such that $x_k f \in L^2(\mathbb{R}^N;\mathbb{C})$ ( $1\leq k \leq n$ ), then $\hat f$ is weakly differentiable with respect to $\xi_k$ and $\partial_k \hat f = \widehat{-i x_k f}$.
(iv) Conversely, if $f \in L^2(\mathbb{R}^N;\mathbb{C})$ such that its weak derivative $\partial_k f \in L^2(\mathbb{R}^N;\mathbb{C})$, then $\widehat{\partial_k f}= i \xi_k \widehat{f}$.
I have some difficulty justifying the above statement from a density argument, especially from (ii) to (iv).
Regarding (iii), I approach $f$ by density in $L^1 \cap L^2$ and I use the following approximation by truncation $f_p(x) = f(x) 1_{[-p,p]^n}(x) 1_{[0,p]}(|f(x)|)$ which converges strongly toward $f$ in $L^2$. Having this explicit approximation is useful so I can verify $x_k f_p \in L^1$ and use (i). Then all I have to do is to study for some $\varphi \in C^{\infty}_c(\mathbb{R}^n)$ $$\left| \int_{\mathbb{R}^n} \hat{f}(\xi) \partial_k \varphi(\xi) \, \mathrm{d} \xi - \int_{\mathbb{R}^n} \widehat{-i x_k f}(\xi) \varphi(\xi) \, \mathrm{d} \xi\right|.$$ Then one just introduces the same quantities replacing $f$ by $f_p$ (using (i)) and pass to the limit $p \rightarrow +\infty$.
My biggest concern is for item (iv). Using the same approximation $f_p(x) = f(x) 1_{[-p,p]^n}(x) 1_{[0,p]}(|f(x)|)$ sounds like a bad idea cause we need to verify that its weak derivatives $\partial f_p$ is in $L^2$ which looks false. I thought of using other approximations like a smooth one...
Do you have any thoughts on this ? I also know that properties (iii) and (iv) can be proved using Schwartz space but that is not my wish here as I only want to deal with $L^1/L^2$ Fourier theory.