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Potential Type Operators On Weighted Variable Exponent Lebesgue Spaces

Vladimir Samuilovich Rabinovich
Published 2013 · Mathematics

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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.
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