A weight space invariant with respect to a singular linear operator

For the singular operator
$$ S_u=\int_a^b\frac{K(x,s)u(s)}{s-x}\,ds $$
invariant weight spaces $\lambda_{\alpha,p}^\beta$ ($u(x)\in\lambda_{\alpha,p}^\beta$ if $1^0$. $u(x)\rho(x)\in H_\beta^0$, $2^0$. $\|u\|_{L_p(\rho_0)}<\infty$, $\rho(x)=(x-a)(b-x)^{1+\beta}$, $\rho_0(x)-(b-x)^{\alpha(p-1)}$, $0<\alpha$, $\beta<1$, $p>1$, $H_\beta^0$ is a Hölder space. Multiplicative inequalities of the type of Kh. Sh. Mukhtarov are also obtained.