Matsugu, Yasuo; Ueki, Sei-ichiro Isometries of weighted Bergman-Privalov spaces on the unit ball of \(\mathbb{C}^n\). (English) Zbl 1027.32012 J. Math. Soc. Japan 54, No. 2, 341-347 (2002). Summary: Let \(B\) denote the unit ball in \(\mathbb{C}^n\), and \(\nu\) the normalized Lebesgue measure on \(B\). For \(\alpha>-1\), define \(d\nu_\alpha(z)= \Gamma(n+ \alpha+1)/\{\Gamma(n+ 1)\Gamma(\alpha+ 1)\}(1-|z|^2)^\alpha d\nu(z)\), \(z\in B\). Let \(H(B)\) denote the space of holomorphic functions in \(B\). For \(p\geq 1\), define \[ (AN)^p(\nu_\alpha)= \Biggl\{f\in H(B):\|f\|\equiv \Biggl[\int_B\{\log(1+|f|)\}^p d\nu_\alpha\Biggr]^{1/p}< \infty\Biggr\}. \] \((AN)^p(\nu_\alpha)\) is an \(F\)-space with respect to the metric \(\rho(f,g)\equiv\|f-g\|\).We prove that every linear isometry \(T\) of \((AN)^p(\nu_\alpha)\) into itself is of the form \(Tf= c(f\circ\psi)\) for all \(f\in (AN)^p(\nu_\alpha)\), where \(c\) is a complex number with \(|c|= 1\) and \(\psi\) is a holomorphic self-map of \(B\) which is measure-preserving with respect to the measure \(\nu_\alpha\). Cited in 5 Documents MSC: 32A36 Bergman spaces of functions in several complex variables 32A38 Algebras of holomorphic functions of several complex variables 46E10 Topological linear spaces of continuous, differentiable or analytic functions Keywords:weighted Bergman-Privalov spaces; unit ball in \(\mathbb{C}^n\); \(F\)-space; isometry PDFBibTeX XMLCite \textit{Y. Matsugu} and \textit{S.-i. Ueki}, J. Math. Soc. Japan 54, No. 2, 341--347 (2002; Zbl 1027.32012) Full Text: DOI