Vekua Integral Operators on Riemann Surfaces

On an arbitrary (in general, noncompact) Riemann surface $R$, we study integral operators $\operatorname{T}$ and $\Pi$ analogous to the operators introduced by Vekua in his theory of generalized analytic functions. By way of application, we obtain necessary and sufficient conditions for the solvability of the nonhomogeneous Cauchy–Riemann equation $\overline\partial f=F$ in the class of functions $f$ exhibiting $\Lambda_0$-behavior in the vicinity of the ideal boundary of $R$.