Riemann's boundary value problem with degenerating coefficient and bases of exponents (Q2910421)

The authors consider, in the class \(H^{+}_{p,\omega^{+}} \times _{-1} H^{-}_{p,\omega^{-}}\), the inhomogeneous boundary value problem NEWLINE\[NEWLINE\begin{aligned} F^{+}(\tau) + G(\tau) F^{-}(\tau) &= \widetilde{f}(\mathrm{arg}, \tau),\quad |\tau| = 1,\\ f^{-}(\infty) &= 0, \end{aligned}NEWLINE\]NEWLINE where \(\widetilde{f}(t) = \frac{f(t)}{\rho^{+}(t)}\), \(f\in L_p(-\pi, \pi)\). The coefficient \(G(t)\) is determined by the relation NEWLINE\[NEWLINE G(e^{i t}) = \frac{\rho^{-}(t) A(t)}{\rho^{+}(t) B(t)}, NEWLINE\]NEWLINE with \(A\) and \(B\) being piecewise Hölder continuous functions, and the weight functions \(\omega^{\pm}(t) = \left(\rho^{\pm}(t)\right)^p\) satisfying NEWLINE\[NEWLINE \rho^{\pm}(t) = |t|^{\beta_{0}^{\pm}} |t - \pi|^{\beta_{\pi}^{\pm}} |t + \pi|^{\beta_{-\pi}^{\pm}} \prod_{k=1}{r^{\pm}} |t - t_{k}^{\pm}|^{\beta_{k}^{\pm}}. NEWLINE\]NEWLINE Solvability of the considered problem is shown under usual conditions on parameters. By using this result the authors deduce the basicity of a double system of exponents with degenerated coefficients given by NEWLINE\[NEWLINE \left\{A(t) \rho^{+}(t) e^{i n t}; B(t) \rho^{-}(t) e^{-i k t}\right\}_{n\in {\mathbb Z}_{+}; k\in {\mathbb N}}. NEWLINE\]NEWLINE Reviewer's remark: The paper contains a considerable number of misprints and misleading notations.