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

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We consider Lipschitz smoothness of an arbitray invariant potential U on the unit ball B in $$\mathbb{C}^n $$ . 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 Ω 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 Ω, such that u is Lipschitz smooth on B\Ω.