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One of the characterizing properties of the functions of bounded mean oscillation (BMO) is that their distribution functions have an exponential decay effect, i. e. the famous John-Nirenberg's theorema:i. In 1980, Baernstein® proved that the distribution functions of the non-tangential maximal functions decrease exponentially for a bounded subset of the Nevanlinna class in the unit disk, and as corollaries, he obtained an analytic form of John-Nirenberg's theorem with a weaker integrability assumption and pointed out that in the analytic category BMO is equivalent to BMO of logarithmic type. Long Ruilin and Yang LeC3] obtained similar results to Baernstein's theorem in the ^-dimensional real and complex ball by showing that BMO(10g)* = BMO for spaces of homogeneous type. In this paper, we try to generalize a series of the famous Baernstein's results for the unit disk to the unit ball with respect to different topological structures applying Rudin's function theory in the unit ball of CM. In order to lead to the discussion, we define a class of point sets in the ball in §2, where we point out that there is a useful geometric property of the intersections of this class of sets and the admissible domains Da (Q defined by Koranyi rs:i. The key part of this paper will be found in §3, where the decay characterizations will be studied for a bounded subset of a function space larger than the Nevanlinna class, which shall be referred to as H9 class in the present article. Hence the maximal function in the admissible domain

Some classes of functions with exponential decay in the unit ball of {${\bf C}\sp n$}