To the theory of a generalized Cauchy-Riemann system with a singular point
The author studies the first order complex equation with a singular point \[ \partial_{\overline z} W + {A(z) \over | z |} W + {B(z) \over | z |} \overline W = F(z) \tag{1} \] in \(G\), where \(G\) is a bounded domain with boundary \(\Gamma \in C^{1, \alpha}\) \((0 < \alpha < 1)\) of variables \(z = re^{i \varphi}\) which contains the point \(z = 0\), and \(A(z)\), \(B(z)\), \(F(z) \in L_ q (G)\), \(2 < q < {2 \over \beta}\), \(0 < \beta < 1\). The existence for solutions to equation (1) in the class \[ W^ 1_ q (G) \cap S_{\beta - 1} (G_ 0), \quad 2 < q < 2/ \beta \tag{2} \] is proved, where \(W_ q^ 1(G)\) is the Sobolev space, \(G_ 0 = \{z, | z | < \varepsilon\}\), \(S(G)\) is the space of measurable and essentially bounded functions \(f(z)\) in \(G\), \(S_ r (G)\) is the class of functions \(f(z) = | z |^{-r} f_ 0 (z)\), \(f_ 0 (z) \in S(G)\), \(r \in \mathbb{R}\). Also, the author constructs representations of first and second kind for solutions in the class (2), and obtains representations of the solutions and solvability conditions of the Riemann-Hilbert problem for equation (1).
scientific article; zbMATH DE number 652190