Let $\omega \in \mathcal{S}(\mathbb{R}^{2})$ and define $u(x) = \int_{\mathbb{R}^{2}}\frac{(x-y)^{\perp}}{|x - y|^{2}}\omega(y)dy$, where
$(x-y)^{\perp} = \begin{bmatrix}x_{2} - y_{2}\\-x_{1} + y_{1} \end{bmatrix}$.
Show that $\partial_{x_{1}}u_{1}$ is related to $\omega$ by a singular integral operator.
Idea: I have tried to compute the Fourier transform of $\frac{y^{\perp}}{|y|^{2}}$, but did not see anything dropping out.