Let $f \in L^2(\mathbb{R}^n)$. For a function $h \in L^2(\mathbb{R}^n)$, let $\hat{h}$ denotes it Fourier transform. Suppose that it is known that the function $$g(\xi) = \sum_{j=1}^n \xi_j \hat{f}(\xi)$$
belongs to $L^2(\mathbb{R}^n)$.
I would like to know whether it is true that each function $\xi_j \hat{f}(\xi)$ belongs to $L^2(\mathbb{R}^n)$.
Conceptually, this propostion is asking whether $f$ having a combination of all of its distributional derivatives in $L^2$ implies that $f \in H^1(\mathbb{R}^n)$.
My attempt so far rests on establishing an inequality of the form $$|\xi_k| \le C \left|\sum_{j =1}^n \xi_j \right|, \qquad k =1 ,\dots , n,$$
for some $C > 0$. This will then say that $\|\xi_k \hat{f}\|_{L^2} \le C \|g\|_{L^2}$, and this is the conclusion I want. But so far I am stuck. Could it be that the proposition is not true?
Hints or solutions are greatly appreciated.
The desired result does not hold.
By performing an orthogonal transformation of $\mathbb{R}^n$, your hypothesis is equivalent to asking whether
$$ g\in L^2(\mathbb{R}^n) \wedge \partial_{x^1} g \in L^2(\mathbb{R}^n) \implies g \in H^1(\mathbb{R}^n) $$
which is clearly false.
For an explicit counter example: Let $n = 2$. Let $\chi$ be the characteristic function of the set $[-1,1]\subset \mathbb{R}$ and let $\eta$ be any smooth bump function. Take
$$ f(x_1,x_2) = \chi(x_1 - x_2) \eta(x_1+x_2) $$
then you can check that $f\in L^2$, and $(\xi_1 + \xi_2) \hat{f} \in L^2$, but $f\not\in H^1$.