Smoothness of Subharmonic Functions and Potentials of the Bergman Metric in the Unit Ball

We consider Lipschitz smoothness of an arbitray invariant potential U on the unit ball B in . We establish some Lipschitz estimates for both U and its gradient vector field U with respect to the Bergman metric. These estimates are taken with respect an invariant distance on B and shown to hold outside on open sets with arbitrarily small Hausdorff conttent. We also prove that for an M-subharmonic function u which satisfies Littelwood's integrability condition, there are such open sets , such that u is Lipschitz smooth on B.