The growth conditions for the membership of positive ℳ-subharmonic Hardy functions

On the complex ball, [Formula: see text]-subharmonic Hardy classes and Bergman classes are studied. Conditions equivalent to the membership of the classes are expressed in terms of the volume integrals and Green’s function, which extend known results for holomorphic case. Riesz decomposition theorem and Hardy–Littlewood theorem of [Formula: see text]-subharmonic function on the unit ball are established. Consequences of the results are given as an illustration.