Singular integral equations on a nonrectifiable curve (Q1901961)

The author considers the singular integral equation \[ a(t) \varphi (t) + b(t) S_\Gamma \varphi (t) = f(t) \] where \(\Gamma\) is a nonrectifiable, closed Jordan curve, \(a(t)\), \(b(t)\), \(f(t) \in H_\nu (\Gamma)\), with \(a^2 (t) - b^2 (t) = 1\), and the singular integral \(S_\Gamma \varphi (t)\) is defined in terms of the boundary values of a piecewise-analytical function. For smooth curves \(\Gamma\), the definition of \(S_\Gamma \varphi (t)\) coincides with the familiar formula \[ S_\Gamma \varphi (t) = {1 \over \pi i} \text{ v.p. } \int_\Gamma {\varphi (\tau) \over \tau - t} d \tau. \] Conditions are given for the existence of solutions \(\varphi \in H_\lambda (\Gamma)\). Results on the associated adjoint equation and the corresponding problem involving a suitable restricted open Jordan curve are also presented.