Abstract
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.
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