Geodesic
Discontinuous Riemann boundary problem in weighted spaces
Proceedings of the Yerevan State University. Physical and mathematical sciences, Tome 51 (2017) no. 1, pp. 38-41.

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The Riemann boundary problem in weighted spaces $L^{1}(\rho)$ on $T=\{t, |t|=1\}$, where $\rho(t)={|t-t_{0}|}^{\alpha}$,   $ t_{0}\in T$ and $\alpha>-1$, is investigated. The problem is to find analytic functions $\Phi^{+}(z)$ and $\Phi^{-}(z),\,\,\Phi^{-}(\infty)=0$ defined on the interior and exterior domains of $T$ respectively, such that:  $ \lim\limits_{r\rightarrow 1-0}\|\Phi^{+}(rt)-a(t)\Phi^{-}(r^{-1}t)-f(t)\|_{L^{1}(\rho)}=0, $ where $f\in L^{1}(\rho),\,\,a(t)\in H_{0}(T;t_{1},t_{2},\dots,t_{m})$. The article gives necessary and sufficient conditions for solvability of the problem and with explicit form of the solutions.
Keywords: Riemann boundary problem, weighted spaces, Cauchy type integral, Hölder classes. 
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V. G. Petrosyan. Discontinuous Riemann boundary problem in weighted spaces. Proceedings of the Yerevan State University. Physical and mathematical sciences, Tome 51 (2017) no. 1, pp. 38-41. http://geodesic.mathdoc.fr/item/UZERU_2017_51_1_a6/

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