Chuaqui, M., & Hernandez, R. (2015). Ahlfors-Weill extensions in several complex variables. J. Reine Angew. Math., 698, 161–179.
Abstract: We derive an Ahlfors-Weill type extension for a class of holomorphic mappings defined in the ball B-n, generalizing the formula for Nehari mappings in the disk. The class of mappings holomorphic in the ball is defined in terms of the Schwarzian operator. Convexity relative to the Bergman metric plays an essential role, as well as the concept of a weakly linearly convex domain. The extension outside the ball takes values in the projective dual to C-n, that is, in the set of complex hyperplanes.