On Clifford theorem on open Riemann surface (Q1902768)

Let \(R\) be a closed Riemann surface of genus \(h\), and \(M(D^{- 1})\) be the space of meromorphic functions divisible by the divisor \(D^{- 1}\). Then (1) \(\dim M(D^{- 1})= \min(0, \text{ord } D- h+ 1)\), for \(\text{ord }D\neq 0, 1, 2,\dots, 2h- 2\). The essense of the classical Clifford theorem is (2) \(\dim M(D^{- 1})\leq [\text{ord } D/2]+ 1\), for \(0\leq \text{ord } D\leq 2h- 2\). The author proves that the following generalization of the Clifford's theorem. Theorem. Let \(M(D^{- 1})\) denote a space of meromorphic functions on the Riemann surface \(R\) of genus \(h\), divisible by the divisor \(D^{- 1}\), \(0\leq \text{ord }D\leq 2h- 2\), and possessing a finite Dirichlet integral outside some compact set. Then the dimension of the space \(M(D^{- 1})\) satisfies bound (2) in the cases: a) when \(R\in O_{\text{AD}}\) and \(h< \infty\), b) when \(R\in O_G\) and the divisor \(D\) is being an entire one. We think that this is an important generalization of Clifford's theorem.