A priori estimates for one-dimensional singular integral operators with continuous coefficients

Let the contour $\Gamma$ consist of a finite number of simple closed pairwise nonintersecting curves, satisfying a Lyapunov condition, let $S$ be the operator of singular integration in space $L_p(\Gamma)(1<p<\infty)$, and let $a(t),b(t)\in C(\Gamma)$, $1<p_1<p<\infty$. The necessary and sufficient condition for $A=al+bS$ to be a $\Phi$-operator in space $L_p(\Gamma)$ is that, for all $\varphi\in L_p(\Gamma)$, $\|\varphi\|_p\le\operatorname{const}(\|A\varphi\|_p+\|\varphi\|_{p_1})$, where $\|\varphi\|_p=\|\varphi\|_{L_p(\Gamma)}$.