Báo Cáo Báo cáo hóa học BOUNDARY VALUE PROBLEMS FOR ANALYTIC FUNCTIONS IN THE CLASS OF CAUCHY-TYPE INTEGRALS

BOUNDARY VALUE PROBLEMS FOR ANALYTIC FUNCTIONS IN THE CLASS OF CAUCHY-TYPE INTEGRALS WITH DENSITY IN L p(·) (Γ) V. KOKILASHVILI, V. PAATASHVILI, AND S. SAMKO Received 9 July 2004 We study the Riemann boundary value problem Φ+ (t) = G(t)Φư (t) + g(t), for analytic functions in the class of analytic functions represented by the Cauchy-type integrals with density in the spaces L p(·) (Γ) with variable exponent. We consider both the case when the coefficient G is piecewise continuous and the case when it may be of a more general nature, admitting its oscillation. The explicit formulas for solutions in the variable exponent setting are given. The related singular integral equations in the same setting are also investigated. As an application there is derived some extension of the Szeg¨ -Helson o theorem to the case of variable exponents. 1. Introduction Let Γ be an oriented rectifiable closed simple curve in the