Riemann boundary value problem on fractal arcs and on arcs with infinite length. I (Q1901922)

The author considers the homogeneous Riemann boundary value problem on fractal arcs and on arcs with infinite length. He finds the solution of the problem \(\phi^+(t)= G(t)\phi^-(t)\), \(t\in \Gamma\backslash \{a_1, a_2\}\), with the following restrictions near the ends \(a_j: \phi(z)= O(|z- a_j|^{- \gamma})\), \(\gamma< 1\). He studies this problem for a class of nonrectifiable curves which contains, in particular, the locally rectifiable curves, and for sufficiently smooth coefficient \(G(t)\). In part I of this work the main attention is connected with the gap problem.