On a Riemann Boundary Value Problem in the Half-plane in the Class of Weighted Continuous Functions

Abstract

Let C(ρ) be the class of functions f such that f(x)ρ(x) is continuous on (−∞,+∞). In the upper half-plane of complex plane z we consider the Riemann boundary value problem in the weighted space C(ρ) with \(\rho \left( x \right) = \prod\limits_{k = 1}^m {{{\left| {\frac{{x - {x_k}}}{{x + i}}} \right|}^{{\alpha _k}}}} \), where α_k and x_k are real numbers, k = 1, 2,...,m. The problem is to determine an analytic in the upper and lower half-planes function Φ(z) to satisfy \(\mathop {\lim }\limits_{y \to + 0} {\left\| {{\Phi ^ + }\left( {x + iy} \right) - a\left( x \right){\Phi ^ - }\left( {x - iy} \right) - f\left( x \right)} \right\|_{C\left( \rho \right)}} = 0\), where f ∈ C(ρ), a(x) ∈ C^δ[−A;A] for any A > 0, a(x) ≠ 0, the limit \(\mathop {\lim }\limits_{\left| x \right| \to \infty } a\left( x \right) = a\left( \infty \right)\) exists and |a(x) − a(∞)| < C|x|^-δ for |x| ≥ A > 0. The normal solvability of this problem is established.