We find an exact asymptotic formula for the singular values of the integral operator of the form $\int _{\Omega } T(x,y)k(x-y) \cdot \mathrm{d}y \: L^2 (\Omega )\rightarrow L^2(\Omega)$ ($\Omega \subset \mathbb{R}^m$, a Jordan measurable set) where $k(t) = k_0((t^2_1 + t^2_2 + \ldots t^2_m)^{\frac{m}{2}})$, $k_0 (x) = x^{\alpha -1} L(\tfrac{1}{x})$, $\tfrac{1}{2} - \tfrac{1}{2m}< \alpha < \tfrac{1}{2}$ and $L$ is slowly varying function with some additional properties. The formula is an explicit expression in terms of $L$ and $T$.
In this paper we study integral operators of the form \[ Tf(x)=\int | x-a_1y|^{-\alpha _1}\dots | x-a_my|^{-\alpha _m}f(y)\mathrm{d}y,
\] $\alpha _1+\dots +\alpha _m=n$. We obtain the $L^p(w)$ boundedness for them, and a weighted $(1,1)$ inequality for weights $w$ in $A_p$ satisfying that there exists $c\ge 1$ such that $w( a_ix) \le cw( x)$ for a.e. $x\in \mathbb R^n$, $1\le i\le m$. Moreover, we prove $\Vert Tf\Vert _{{\mathrm BMO}}\le c\Vert f\Vert _\infty $ for a wide family of functions $f\in L^\infty ( \mathbb R^n)$.