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Isometries of weighted Bergman-Privalov spaces on the unit ball of \(\mathbb{C}^n\). (English) Zbl 1027.32012

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 \(\nu_\alpha\).

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
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