The conjugation problem on an infinite set of intervals (Q1105079)

Let E be an infinite set of intervals \((a_ k,b_ k)\), \(0<a_ k<b_ k<a_{k+1}\), \(k=1,2,..\). on the real axis, satisfying the condition: there exists a number \(\alpha\), \(0<\alpha<1/2\) such that \(b_{k+1}\leq \alpha a_ k\), \(a_ k>0\) for all \(k\geq k_ 0\). The function \(\phi(z)\) defined in the plane z is said to belong to K if: 1) it is analytic outside the closure \(\bar E\) of the set E; 2) it is continuously extendable from above \(\phi^+(x)\) and from below \(\phi^-(x)\) for every \(x\in E\); 3) in the neighborhood of the endpoints \(c_ k=a_ k,b_ k\) it satisfies the inequality \[ | \phi (z)| \leq D_ k(z-c_ k)^{-\gamma_ k}, \] which positive constants, which can increase indefinitely as \(k\to +\infty\) and \(0<\gamma_ k<\gamma <1\); (4) it has a finite degree at infinity. The author considers the following version of the jump problem: Find a K- class function \(\phi(z)\), satisfying the boundary value condition \[ \phi^+(x)=G(x)\phi^-(x),\quad x\in E, \] where \(G(x)\in H(E)\), \(\inf | G(x)| \geq \delta >0\), \(x\in E\); \(\phi^{\pm}(x)\) are the limit values on E from above and below respectively. The associated homogeneous problem and the corresponding singular integral equations are considered.