Uniform estimate for the Cauchy-Riemann equations on a hyperbolic Riemann surface

Question

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.

Answer 1

It is too optimistic to obtain uniform estimates on so large class of Riemann surfaces as the hyperbolic ones. According to the book "Riemann Surfaces" by Ahlfors and Sario there do exist Riemann surfaces (in fact domains in $\mathbb C$) that admit nonconstant bounded subharmonic functions (in particular they admit Green's function) but admit no non-constant holomorphic functions. Such a domain can be realized as the complement of a Cantor-like set for example. Now assuming uniform estimates on such a domain one easily comes with a contradiction with the scarcity of holomorphic solutions provided by the Cauchy-Riemann equations.

Concerning the second question, I have no idea. I think it would be very enlightening if You share with us how would one come up with such a conjecture.