Abstract: | In this paper we consider weighted non-tangential and tangential boundary limits of non-negative functions on the unit ball in that are subharmonic with respect to the Laplace-Beltrami operator on . Since the operator is invariant under the group of holomorphic automorphisms of , functions that are subharmonic with respect to are usually referred to as -subharmonic functions. Our main result is as follows: Let be a non-negative -subharmonic function on satisfying for some and some , where is the -invariant measure on . Suppose . Then for a.e. , uniformly as in each , where for ( when ) We also prove that for the only non-negative -subharmonic function satisfying the above integrability criteria is the zero function. |