Asymptotical properties of harmonic and $M$-harmonic functions near the boundary of the unit sphere

On the boundary of the complex $n$-ball, there are two a natural notions of Hausdorff dimension, namely, those related to the Euclidean and the Koranyi metric. It is shown that “Riesz decompositions” relative to these two dimension scales are linked rigidly for the measures that are boundary values of pluriharmonic functions in the ball.