tailieunhanh - BOUNDARY VALUE PROBLEMS FOR ANALYTIC FUNCTIONS IN THE CLASS OF CAUCHY-TYPE INTEGRALS WITH DENSITY IN

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)Φ− (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. | BOUNDARY VALUE PROBLEMS FOR ANALYTIC FUNCTIONS IN THE CLASS OF CAUCHY-TYPE INTEGRALS WITH DENSITY IN LP T V. KOKILASHVILI V. PAATASHVILI AND S. SAMKO Received 9 July 2004 We study the Riemann boundary value problem o i G t - t g t for analytic functions in the class of analytic functions represented by the Cauchy-type integrals with density in the spaces Lp l ĩ 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 Szego-Helson theorem to the case of variable exponents. 1. Introduction Let r be an oriented rectifiable closed simple curve in the complex plane C. We denote by D and D- the bounded and unbounded component of C r respectively. The main goal of the paper is to investigate the Riemann problem find an analytic function ffi on the complex plane cut along r whose boundary values satisfy the conjugacy condition ffi t G t ffi- t g t t e r where G and g are the given functions on r and ffi and ffi- are boundary values of ffi on r from inside and outside r respectively. This problem is also known as the problem of linear conjugation. We seek the solution of in the class of analytic functions represented by the Cauchy-type integral with density in the spaces Lp l ĩ with variable exponent assuming that g belongs to the same class. We consider the cases when the coefficient G is continuous or piecewise continuous as well as the case of oscillating coefficient. The solvability conditions are derived and in all the cases of solvability the explicit formulas are given. The related boundary singular integral equations in Lp r are treated. The solution of the boundary value problem BVP allows us to obtain the weight .