The Riemann–Hilbert Problem in a Domain with Piecewise Smooth Boundaries in Weight Classes of Cauchy Type Integrals with a Density from Variable Exponent Lebesgue Spaces

Abstract

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.

Singular integral operators on Nakano spaces with weights having finite sets of discontinuities

“…In particular, this implies that ψ p/p * ∈ A p * (Γ) if and only if ψ p(t)/p * ∈ A p * (Γ). Thus, inequalities (19) imply that ψ p/p * ∈ A p * (Γ). Applying Theorem 3.4, we finally conclude that the maximal operator M is bounded on L p(•) (Γ, ψ).…”

“…Fredholmness of one-dimensional singular integral operators on Nakano spaces (variable Lebesgue spaces) over sufficiently smooth curves was studied for the first time by Kokilashvili and S. Samko [25]. The closely related Riemann-Hilbert boundary value problem in weighted classes of Cauchy type integrals with density in L p(•) (Γ) was considered by Kokilashvili, Paatashvili, and S. Samko [18,19,20,22]. The author [12] found a Fredholm criterion for an arbitrary operator in the Banach algebra of one-dimensional singular integral operators with piecewise continuous coefficients acting on Nakano spaces L p(•) (Γ, w) with radial oscillating weights (3) over so-called logarithmic Carleson curves.…”

Double bases from generalized Faber polynomials with complex-valued coefficients in weighted Lebesgue spaces

“…These problems were studied in [31]. Note that similar problems in various formulation were studied in [32,33,34,35,36].…”

The dirichlet problem for harmonic functions from variable exponent smirnov classes in domains with piecewise smooth boundary

“…In recent years, boundary value problems were considered in Lebesgue spaces with a variable exponent (cf., for example, [6]- [8] and other references). Representing the sought for functions by the Cauchy type integral with a density from variable exponent Lebesgue spaces, we investigated the following boundary value problems of the theory of functions of a complex variable: the Riemann, Riemann-Hilbert, and Riemann-Hilbert-Poincaré boundary value problems and also the Dirichlet problem in the class of harmonic functions which are the real parts of above-mentioned Cauchy type integrals (cf., for example, [9]- [12]).…”