I have been trying to find the answer to this question in the literature, but have not succeeded. The question is as follows.

Suppose $X$ is a Riemann surface that admits a green's function (i.e. it has a bounded subharmonic function). Is there a solution of the inhomogeneous Cauchy-Riemann equations $\bar \partial u = \theta$ with uniform estimates? If not, what if we assume the injectivity radius of the Poincare metric is bounded below by a positive constant?

I know the answer is affirmative if $X$ is a so-called finite Riemann surface, i.e., a compact Riemann surface with boundary consisting of finitely many smooth Jordan curves.