On the solvability of Riemann-Hilbert problem in the weighted Smirnov classes
Let $D^+\subset\mathbb{C}$ be a bounded domain with the boundary $\Gamma=\partial D^+$, a curve of bounded rotation without cusps, $D^-$ be its exterior domain, which contains infinity. The main problem under consideration is the following homogeneous Riemann-Hilbert problem $$ A(\xi)F^+(\xi)+B(\xi)F^-(\xi)=0 $$ a.e. on the boundary $\Gamma$. Here, $F^+(\xi)$ \ $(F^-(\xi))$ is a non-tangential value of a function $F^+$ \ $(F^-)$ from the weighted Smirnov class $E_{p,\rho}(D^+)$ ($E^{(m)}_{p,\rho}(D^-)$). The functions from the latter class have the order at most $m$ at infinity. Concerning the coefficients $A=\vert A\vert e^{i\alpha}$, $B=\vert B\vert e^{i\beta}$ the authors make the assumptions \begin{itemize} \item[(i)] $\vert A\vert ^{\pm}$, $\vert B\vert ^{\pm}$ belong to $L^\infty(\Gamma)$; \item[(ii)] $\alpha$ and $\beta$ are piecewise-continuous on $\Gamma$. \end{itemize} The main result of the paper states that the general solution of this problem admits the representation $F=ZP_m$, where $Z$ is a canonical solution of the problem, and $P_m$ is an arbitrary polynomial of degree at most $m$. In particular, in the case $F(\infty)=0$ the problem has only trivial solution.\par The non-homogeneous Riemann-Hilbert problem is also studied, and its general solution is obtained.
scientific article; zbMATH DE number 7011193