On Carleman–Vekua Equations with Nonfinitely Supported Coefficients on a Noncompact Riemann Surface

In this paper, we study equations of the form $\partial f+Af+B\bar f=G$, on an arbitrary noncompact Riemann surface $R$, where $A$, $B$, and $G$ are given square-integrable linear differentials of genus $(0,1)$ satisfying certain additional conditions. Necessary and sufficient conditions for the solvability of the above equation are proved for the class of functions with $\Lambda_0$-behavior in the neighborhood of the ideal boundary of the surface $R$; the index of the equation is also calculated.