Consider the function define by the following Fourier transform:
$$ f(x):= \int \frac{d^nk}{(2\pi)^n} (|k|^2 + m^2)^{-(n+2)/2} \ln(|k|^2 + m^2)^{-\alpha} e^{i k\cdot x} $$
where $x,k\in \mathbb{R}^n$ and $\alpha\geq 0$. When $\alpha=0$, $f(x)$ is the standard Bessel potential and consequently can be expressed as $$ f(x)= \frac{m^{-2}}{(4\pi)^{n/2} \Gamma\left(\frac{d+2}{2}\right)} (mx) K_1(mx) $$ where $K_1$ is a modified Bessel function. Consequently, for small $|x|$ one can use the asymptotics $zK_1(z)$ to find the behavior or $f(x)$.
My question is: when $\alpha>0$, what is the the leading order behavior for $|x|\ll 1$ for $f(x)$?