## 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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