Singular integral equations and the Riemann boundary value problem with infinite index in the space~$L_p(\Gamma,\omega)$

S. M. Grudskii

Izvestiya. Mathematics , Tome 26 (1986) no. 1, pp. 53-76

Résumé

The Riemann boundary value problem $$ \varphi^+(t)-a(t)\varphi^-(t)= f(t),\qquad t\in\Gamma, $$ is considered on a simple closed piecewise smooth contour $\Gamma$ in the space $L_p(\Gamma,\omega)$, along with the corresponding singular integral operator $$ A_{a,\Gamma}=P_\Gamma^+-a(t)P_\Gamma^- $$ with a bounded coefficient $a(t)$ bounded away from zero and having finitely many discontinuities of the second kind that are vorticity points of power type. A theory of one-sided invertibility of $A_{a,\Gamma}$ is constructed, the spaces $\operatorname{Ker}A_{a,\Gamma}$ and $\operatorname{Im}A_{a,\Gamma}$ are described, and a construction is given for the inverse operators. Bibliography: 31 titles.