Given nice maps $f,g$ from $\mathbb{R}^n$ to $\mathbb{R}$, is it true that $\int |\xi|^2\overline{\hat{f}}\hat{g}\, d\xi=\int \overline{Df}Dg\, d\xi$? Using $i\xi_j\hat{f}(\xi) = \widehat{D_jf(\xi)}$, I tried
\begin{align*} |\xi|^2\overline{\hat{f}}\hat{g} &= \sum_j \xi_j^2\overline{\hat{f}}\hat{g}\\ &= \sum_j\overline{i\xi_j\hat{f}}\ i\xi_j\hat{g}\\ &= \sum_j\overline{\widehat{D_jf}}\widehat{D_jg} \end{align*}
Hence, integrating yields $$\sum_j(\widehat{D_jf},\widehat{D_jg})=\sum_j(D_jf,D_jg),$$ by Parseval, which I don't think equals $\int \overline{Df}Dg\, d\xi$. Any help?
Edit: The reason why I'm asking is that I have been given two expressions for the inner product on the Sobolev space $H^1$ (which should be equal up to a constant): $$\int \overline{\widehat{f}(\xi)}\widehat{g}(\xi)(1+|\xi|^2)\, d\xi\quad \text{and}\quad \int \overline{Df(\xi)}Dg(\xi)+\overline{f(\xi)}g(\xi)\, d\xi$$