Characterizations of spaces of holomorphic functions in the ball

Abstract

Let f be holomorphic in the unit ball of Cⁿ. Several equivalent criteria for f to belong to the Hardy space H^p as well as the weighted Bergman space A_q^p, 0<p<∞, q>0, of the ball are established. In the one variable case, some of the above conditions reduce to those of Yamashita, characterizing Hardy spaces of the unit disk. In addition, various identities for the norm of f, in terms of a certain integrated counting function and certain Lusin characteristics, are obtained.