Geodesic
On the $\bar \partial $-problem with $L^2$-estimates on a Riemann surface
Trudy Matematicheskogo Instituta imeni V.A. Steklova, Modern problems of mathematics, mechanics, and mathematical physics, Tome 290 (2015), pp. 280-292.

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The $L^2$-estimates obtained by Hörmander for the solutions to the $\bar \partial $-problem are specified and complemented in the simplest one-dimensional case of Riemann surfaces. 
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E. M. Chirka. On the $\bar \partial $-problem with $L^2$-estimates on a Riemann surface. Trudy Matematicheskogo Instituta imeni V.A. Steklova, Modern problems of mathematics, mechanics, and mathematical physics, Tome 290 (2015), pp. 280-292. http://geodesic.mathdoc.fr/item/TM_2015_290_a22/

[1] Ganning R.C., Narasimhan R., “Immersion of open Riemann surfaces”, Math. Ann., 174 (1967), 103–108 | DOI | MR

[2] Hedenmalm H., On Hörmander's solution of the $\bar \partial $-equation, E-print, 2013, arXiv: 1311.2020 [math.AP]

[3] Hörmander L., Notions of convexity, Prog. Math., 127, Birkhäuser, Boston, MA, 1994 | MR | Zbl

[4] Napier T., Ramachandran M., An introduction to Riemann surfaces, Springer, New York, 2011 | MR | Zbl