Some new characterizations of Dirichlet type spaces on the unit ball of \(\mathbb C^{n}\) (Q852825)

By a Dirichlet type space on the unit ball of \(\mathbb C^n\), denoted by \(D_p\), the author means a Hilbert space consisting of holomorphic functions whose Taylor coefficients are square summable with respect to a natural weight depending on the parameter \(p\). As the parameter \(p\) varies, these Dirichlet type spaces interpolate the well-known function spaces such as the Bergman space (\(p=-1\)), the Hardy space (\(p=0\)) and the classical Dirichlet space (\(p=n\)). In this paper the author provides several characterizations of the Dirichlet type spaces in terms of integrals involving (i) invariant gradient, (ii) invariant mean oscillation, and (iii) mean oscillation in the Bergman metric. However, the parameter range is restricted to \(-1\leq p<n\). These characterizations extend an earlier one obtained by \textit{P. Hu} and \textit{W. Zhang} [J. Math. Anal. Appl. 259, 453--461 (2001; Zbl 1011.46021)] As an application, the author shows that the spaces \(D_p\) with \(-1\leq p<n\) are automorphism invariant.