I'm looking to find the Fourier transform of the distribution $$|x|^{-n+q+2}|Tx|^{-1}$$ where $-1 < q < n-2$, $T$ is a linear transformation, and $|\cdot|$ is the Euclidian norm on $\mathbb{R}^n$.
My first question is whether I can use the convolution theorem, which would say that the distribution I want is:
$$C_1|x|^{-q-2} * C_2|Tx|^{-n+1},$$
where the $C_i$ are positive constants. Here, the question is how to evaluate this convolution, or determine whether it exists. Neither of these are Schwartz, and as distributions they are only locally integrable. Do I need to straightforwardly show that the integral making up the convolution is finite? Is there something I can do here with approximate identities? Thank you.