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

• Published in: Journal of contemporary mathematical analysis Vol. 54; no. 2; pp. 79 - 89
• Main Authors: Hayrapetyan, H. M., Aghekyan, S. A.
• Format: Journal Article
• Language: English
• Published: Moscow Pleiades Publishing 01.03.2019; Springer Nature B.V
• Subjects: Boundary value problems; Continuity (mathematics); Mathematics; Mathematics and Statistics; Real and Complex Analysis; Real numbers; weighted space; Cauchy type integral; Loran expansion; Riemann problem; factorization; 35J25
• ISSN: 1068-3623; 1934-9416
• DOI: 10.3103/S1068362319020043

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 ρ ( x ) = ∏ k = 1 m | x − x k x + i | α 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 lim y → + 0 ‖ Φ + ( x + i y ) − a ( x ) Φ − ( x − i y ) − f ( x ) ‖ C ( ρ ) = 0 , where f ∈ C ( ρ ), a ( x ) ∈ C δ [− A ; A ] for any A > 0, a ( x ) ≠ 0, the limit lim | x | → ∞ a ( x ) = a ( ∞ ) exists and | a ( x ) − a (∞)| < C | x | -δ for | x | ≥ A > 0. The normal solvability of this problem is established.