Nonclassical weighted norm estimates for some Calderón–Zygmund operators on the plane

Let $\mu$ be a Borel measure with a compact support $F\subset\mathbb C$, $\rho$ be the distance from the set $F$;
$$ A_K(f)(z)=\int_FK(\zeta,z)f(\zeta)\,dm(\zeta),\qquad z\in\mathbb C\setminus F, $$
where $K(\zeta,z)=(\zeta-z)^{-2}$ or $K(\zeta,z)=(|\zeta-z|(\zeta-z))^{-1}$ and $m$ is the Lebesque measure. Let $\psi\colon(0,+\infty)\to\mathbb R_+$ be a nondecreasing positive function, $\Phi(z)=\psi(\rho(z))\rho(z)$, $z\in\mathbb C\setminus F$.
We prove that under some additional assumptions on p, the operator $A_K$ is bounded from $L^2(\mu)$ to $L^2(\Phi m)$ if and only if
$$ \int^1_0\frac{\psi(t)}t\,dt+\int_1^{+\infty}\frac{\psi(t)}{t^2}\,dt<+\infty. $$
This means that the interference effect is not observed in such situations. Bibliography: 4 titles.