# Potential Type Operators On Weighted Variable Exponent Lebesgue Spaces

We consider double-layer potential type operators acting in weighted variable exponent Lebesgue space $$L^{p(.)}(\Gamma,w)$$ on some composed curves with oscillating singularities. We obtain a Fredholm criterion for operators $$A=aI+bD_{g.\Gamma}:L^{p(.)}(\Gamma,w)\rightarrow L^{p(.)}(\Gamma,w) \; {\rm where}{D_{g,\Gamma}}$$ is the operator of the form $${D_{g,\Gamma}{u(t)}}=\frac{1}{\pi}\int_{\Gamma}\frac{g(t,\tau)(\nu(\tau),\tau-t)u(\tau)dl_{t}}{|t-\tau|^{2}},t \in \Gamma$$ $$\nu(\tau)$$ is the inward unit normal vector to Γ at the point $$\tau \in \Gamma \setminus \mathcal{F},dl_{\tau}$$ is the oriented Lebesgue measure on $$\tau ,\mathcal{F}$$ is the set of the nodes, $$a,b:\Gamma\rightarrow\mathbb{C},g:\Gamma\times\Gamma \rightarrow \mathbb{C}$$ are a bounded functions with oscillating discontinuities at the nodes only.