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. 
V.P. Havin, A.H. Nersessian, J. Ortega-Cerda