Bergman-Carleson measures and Bloch functions on strongly pseudoconvex domains (Q2712242)

Let \(\Omega\) be a strongly pseudoconvex domain in \(\mathbb{C}^n\) with \(C^\infty\) boundary \(\partial\Omega\). Denote by \(F(z, \xi)\) the infinitesimal Kobayashi metric at \(z\in \Omega\) in the direction \(\xi \in \mathbb{C}^n\), that is, NEWLINE\[NEWLINE F_k(z,\xi) = \inf\{ \alpha >0: \exists u\in \triangle(\Omega) \;\text{with} u(0)=z, \;\text{and} u'(0)=\xi/\alpha \}, NEWLINE\]NEWLINE where \(\triangle(\Omega)\) denotes the set of all holomorphic mappings from the open unit disc \(\triangle\) to \(\Omega\). For a smooth function \(f\) on \(\Omega\), its modulus of the covariant derivative \(Q(f)\) at \(z\in \Omega\) is defined by NEWLINE\[NEWLINE Q(f)(z) = \sup_{0\neq\xi\in\mathbb{C}^n} \frac{1}{F(z,\xi)} \left(\Big|\sum_{j=1}^n \frac{\partial f}{\partial z_j}(z)\xi_j \Big|+ \Big|\sum_{j=1}^n \frac{\partial f}{\partial \bar z_j}(z)\bar\xi_j\Big|\right). NEWLINE\]NEWLINE Considering the following measure NEWLINE\[NEWLINE d\mu_f^p(z) = Q(f)(z)^p dV(z), NEWLINE\]NEWLINE for \(0<p<\infty\), where \(dV\) is the \(2n\) dimensional Lebesgue measure on \(\mathbb{C}^n\). NEWLINENEWLINENEWLINEThe main result of the paper is the following theorem. Suppose \(f\) is a pluriharmonic function on \(\Omega\). NEWLINENEWLINENEWLINE(A) The following statements are mutually equivalent: (i) \(f\) is a pluriharmonic Bloch function on \(\Omega\). (ii) \(\mu_f^p\) is a Bergman-Carleson measure on \(\Omega\) for every \(0<p<\infty\). (iii) \(\mu_f^p\) is a Bergman-Carleson measure on \(\Omega\) for some \(0<p<\infty\). NEWLINENEWLINENEWLINE(B) The following statements are mutually equivalent: (i) \(f\) is a pluriharmonic little Bloch function on \(\Omega\). (ii) \(\mu_f^p\) is a vanishing Bergman-Carleson measure on \(\Omega\) for every \(0<p<\infty\). (iii) \(\mu_f^p\) is a vanishing Bergman-Carleson measure on \(\Omega\) for some \(0<p<\infty\). NEWLINENEWLINENEWLINECase \(p=2\) of (A) can be found in [\textit{J. S. Choa, H. O. Kim} and \textit{Y. Y. Park}, Bull. Korean Math. Soc. 29, 285-293 (1992; Zbl 0761.32004)]. Case (B) for the unit disc can be found in [\textit{J. Xiao}, Ann. Acad. Sci. Fenn., Ser. A I Math. 19, 35-46 (1994; Zbl 0816.30025) and \textit{J. Xiao} and \textit{L. Zhong}, Complex Var. Theory Appl. 27, 175-184 (1995; Zbl 0845.30023)].NEWLINENEWLINEFor the entire collection see [Zbl 0957.00034].