Uniform estimates in the Poincare-Aronszajn Theorem on the separation of singularities of analytic functions

We study the possibility of splitting any bounded analytic function with singularities in a closed set E union F as a sum of two bounded analytic functions with singularities in E and F respectively. We obtain some results under geometric restrictions on the sets E and F and we provide some examples showing the sharpness of the positive results.