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 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 Szegö-Helson theorem to the case of variable exponents.

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We consider the Riemann-Hilbert problem in the following setting: find a function φ ∈ K p( · ) (D; ω) whose boundary values φ + (t) satisfy the condition Re[(a(t) + ib(t))φ + (t)] = c(t) a.e. on Γ. Here D is a simply connected domain bounded by a simple closed curve Γ, andis a weight function and (KΓϕ)(z) is a Cauchy type integral whose density ϕ is integrable with a variable exponent p(t). It is assumed that Γ is a piecewiseLyapunov curve without zero angles, ω(z) is an arbitrary power function and p(t) satisfies the Log-Hölder condition. The solvability conditions are established and solutions are constructed. These solutions largely depend on the coefficients a, b, c, the weight ω, on the values of p(t) at the angular points of Γ and on the values of angles at these points. Mathematics Subject Classification (2000). Primary 47B38, 30E20, 30E25; Secondary 42B20, 45P05.

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The Riemann–Hilbert problem for an analytic function is solved in weighted classes of Cauchy type integrals in a simply connected domain not containing 𝑧 = ∞ and having a density from variable exponent Lebesgue spaces. It is assumed that the domain boundary is a piecewise smooth curve. The solvability conditions are established and solutions are constructed. The solution is found to essentially depend on the coefficients from the boundary condition, the weight, space exponent values at the angular points of the boundary curve and also on the angle values. The non-Fredholmian case is investigated. An application of the obtained results to the Neumann problem is given.