Characterizations of certain holomorphic function spaces in the unit ball of \(\mathbb{C}^n\) (Q2706875)

This paper investigates a characterization of the Bergman space \(L_{a}^{p}\), Bloch space \(\mathcal B\) and Hardy space \(H^{p}\) of holomorphic functions in the unit ball \(B\) of \(\mathbb{C}^{n}\). One of the main results is about the Bergman space as follows: For \(s>0\) and \(p>\max (0,2-s)\), \(f \in L_{a}^{p}(B)\) is equivalent to NEWLINE\[NEWLINEG_{p,s}(f)=\int_{B}\left|\widetilde{\nabla} f \right|^2 (z)\left|f(z) \right|^{p-2} \frac{(1-\left|z \right|^2)}{n+1}\left|z \right|^{2n-s} d \lambda (z)^{\frac{1}{p}} < \infty NEWLINE\]NEWLINE where \(\widetilde {\nabla}\) and \(\lambda\) are the invariant gradient and invariant measure on \(B\) respectively, and further if \(f(0)=0\), then \(\|f \|_{L_{a}^{p}} \cong G_{p,s}(f)\). A corresponding characterization holds for functions of the Bloch space \(\mathcal B (B)\) which solves a conjecture stated by \textit{C. Ouyang, W. Yang} and \textit{R. Zhao} [Trans. Am. Math. Soc. 347, No. 11, 4301-4313 (1995; Zbl 0849.32005)]. Similar results for the Hardy space and BMOA space are also obtained.