The Riemann-Hilbert problem for the generalized Cauchy-Riemann equation with a singularity (Q2782703)

The authors study the following Riemann-Hilbert problem: NEWLINE\[NEWLINE\partial_{\overline z}w +{a\over 2\overline z}w+ {b\over 2\overline z} \overline w=0\;(a,b\in \mathbb{C})NEWLINE\]NEWLINE with the boundary condition \(\text{Re}[z^{-m}w] =g(z)\), \(m\in Z\) where \(g\) is a given function. The solution is searched in the space \(L_{a_1,\infty} \cap C_{\overline z}(G\setminus \{0\})\cap C(\overline G \setminus\{0\}\) where \(L_{a_1,\infty}= \{f:|f(z)||z|^{\text{Re}a}\in L_\infty (G)\}\). The investigation of this boundary value problem based on a common paper of \textit{A. Timofew} together with \textit{M. Reissig} [Appl. Analysis 73, 187-199 (1999; Zbl 1039.30025] where the problem was treated in the case of \(m=0\).