I'm trying to better understand a convolution operator of the form $T_i f = K * f$ in $\mathbb{R}^3$, where $K(x) = \frac{x_i/|x|}{|x|}$. Without the $x_i/|x|$ in the numerator, this would be precisely the inverse Laplacian (modulo constants). It seems to almost behave similarly to the inverse Laplacian too, since the corresponding Fourier multiplier $\hat{K}$ will be homogeneous of degree $-2$.
Anyways, I want to explicitly compute $\hat{K}$, and my approach was to first notice that $\Delta K = -2\frac{x_i/|x|}{|x|^3}$ away from the origin, so (this is the step I'm unsure of) $\Delta K$ is the kernel for the Riesz transform, which means that $$\widehat{\Delta K} = -|\xi|^2\hat{K} = -2C\frac{\xi_i}{|\xi|} \implies \hat{K} = 2C\frac{\xi_i/|\xi|}{|\xi|^2}$$ This makes some amount of sense, since we know a priori that $\hat{K}$ is homogeneous of degree $-2$, but my main question is whether the distribution $\Delta K$ actually behaves like the Riesz transform, or whether there's some extra artifacts like a Dirac mass or its derivatives due to the singularity at the origin.