Geodesic
A~priori estimates for one-dimensional singular integral operators with continuous coefficients
Matematičeskie zametki, Tome 17 (1975) no. 6, pp. 851-856.

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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 $L_p(\Gamma)(1$, and let $a(t),b(t)\in C(\Gamma)$, $1$. 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 $\|\varphi\|_p=\|\varphi\|_{L_p(\Gamma)}$. 
@article{MZM_1975_17_6_a2,
     author = {V. S. Pilidi},
     title = {A~priori estimates for one-dimensional singular integral operators with continuous coefficients},
     journal = {Matemati\v{c}eskie zametki},
     pages = {851--856},
     publisher = {mathdoc},
     volume = {17},
     number = {6},
     year = {1975},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_1975_17_6_a2/}
}
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V. S. Pilidi. A~priori estimates for one-dimensional singular integral operators with continuous coefficients. Matematičeskie zametki, Tome 17 (1975) no. 6, pp. 851-856. http://geodesic.mathdoc.fr/item/MZM_1975_17_6_a2/