Characterizations of spaces of holomorphic functions in the ball (Q1061904)

Several characterizations are given for Hardy spaces \(H^ p\) \((0<p<\infty)\) and weighted Bergman spaces \(A^ p_ q\) \((0<p<\infty\), \(q>0)\) of holomorphic functions in the unit ball of \({\mathbb{C}}^ n\). These include the characterization of \(H^ p\) as the space of all holomorphic functions f such that the volume integral \(\int R \bar R| f|^ p(1-| z|^ 2)dw\) is finite, where \(R=\Sigma z_ j\partial_ j\) is the radial derivative operator. (This result extends a result of Yamashita in the unit disk). Additional characterizations are given in terms of integrated counting functions and Lusin characteristics.