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On the defect numbers of the Kveselava-Vekua operator with discontinuous derivative of the shift. (English. Russian original) Zbl 0770.45002

Sov. Math., Dokl. 43, No. 3, 633-638 (1991); translation from Dokl. Akad. Nauk SSSR 318, No. 1, 11-16 (1991).
The authors consider the Kveselava-Vekua operator (1) \(K=WP_ ++GP_ -\) in the space \(L_ p(\Gamma,\rho)\) with weight \(\rho(t)=| t-t_ 0|^{\beta_ 0}| t-t_ 1|^{\beta_ 1}\), \(p^{- 1}<\beta_ j<1-p^{-1}\), \(1<p<\infty\), where \(W\varphi=\varphi\circ\alpha\), \(P_ \pm={1\over 2}(I\pm S)\), \(G(t)\in C(\Gamma)\), \(\Gamma=[t_ 0,t_ 1]\) being an open oriented Lyapunov curve, and \(\alpha\) an orientation-preserving \(H\)-smooth diffeomorphism of \(\Gamma\) onto itself.
First, the authors prove a new theorem on conformal gluing. With the help of this theorem they reduce the operator (1) to a singular integral operator \(\tilde K=P_ ++\tilde GP_ -\) (with Cauchy kernel), and obtain sufficient and necessary conditions for \(K\) to be a Fredholm operator and a condition satisfied for \(K\) when \(K\) and \(\tilde K\) are Fredholm operator simultaneously.

MSC:

45E05 Integral equations with kernels of Cauchy type
45P05 Integral operators
47A53 (Semi-) Fredholm operators; index theories
47G10 Integral operators
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