Potential type operators on weighted variable exponent lebesgue spaces

We consider double-layer potential type operators acting in weighted variable exponent Lebesgue space Lp(.)(Γ,w) on some composed curves with oscillating singularities. We obtain a Fredholm criterion for operators (formula presented) where D_g,Γ is the operator of the form (formula presented) ν(τ) is the inward unit normal vector to Γ at the point (formula presented) is the oriented Lebesgue measure on τ,F is the set of the nodes, a,b:Γ→C,g:Γ×Γ→C are a bounded functions with oscillating discontinuities at the nodes only. We give applications of such operators to the Dirichlet and Neumann problems with boundary function in Lp(.)(Γ,w) for domains with boundaries having a finite set of oscillating singularities.