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Isometries of a Bergman-Privalov-type space on the unit ball. (English) Zbl 1178.32004

Summary: We introduce a new space \(AN_{\log,\alpha}(\mathbb B)\) consisting of all holomorphic functions on the unit ball \(\mathbb B\subset\mathbb C^N\) such that \(\|f\|_{AN_{\log,\alpha}}:= \int_{\mathbb B}\varphi_e(\ln(1+|f(z)|))\, dV_\alpha(z)<\infty\), where \(\alpha>-1\), \(dV_\alpha(z)= c_{\alpha,n}(1-|z|^2)^\alpha\, dV(z)\) (\(dV(z)\) is the normalized Lebesgue volume measure on \(\mathbb B\), and \(c_{\alpha,n}\) is a normalization constant, that is, \(V_\alpha(\mathbb B)=1\)), and \(\varphi_e(t)= t\ln(e+t)\) for \(t\in [-,\infty)\). Some basic properties of this space are presented. Among other results, we prove that \(AN_{\log,\alpha}(\mathbb B)\) with the metric \(d(f,g)= \|f-g\|_{AN_{\log,\alpha}}\) is an \(F\)-algebra with respect to pointwise addition and multiplication. We also prove that every linear isometry \(T\) of \(AN_{\log,\alpha}(\mathbb B)\) into itself has the form \(Tf=c(f\circ\psi)\) for some \(c\in\mathbb C\) such that \(|c|=1\) and some \(\psi\) which is a holomorphic self-map of \(\mathbb B\) satisfying a measure-preserving property with respect to the measure \(dV_\alpha\). As a consequence of this result, we obtain a complete characterization of all linear bijective isometries of \(AN_{\log,\alpha}(\mathbb B)\).

MSC:

32A37 Other spaces of holomorphic functions of several complex variables (e.g., bounded mean oscillation (BMOA), vanishing mean oscillation (VMOA))
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