Stevo Stevic
Isometries on the Bergman-Privalov class on the unit ball
Bergman-Privalov class $AN_{\alpha}(\mathbb{B})$ consists of all holomorphic functions on the unit ball $\mathbb{B}\subset\mathbb{C}^n$ such that
$$\|f\|_{AN_{\alpha}}:=\int_{\mathbb{B}}\ln(1+|f(z)|)dV_{\alpha}(z)<\infty,$$
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 the normalization constant, that is, $V_{\alpha}(\mathbb{B})=1$). Under a mild condition, we characterize surjective isometries (not necessarily linear) on $AN_{\alpha}(\mathbb{B}),$ and prove that $T$ is surjective multiplicative isometry (not necessarily linear) on $AN_{\alpha}(\mathbb{B})$ if and only if it has the form
$Tf=f\circ\psi$ or $Tf=\overline{f\circ\overline{\psi}},$
for every $f\in AN_{\alpha}(\mathbb{B}),$ where $\psi$ is a unitary transformation of the unit ball. The corresponding results for the case of the Bergman-Privalov space on the unit polydisk $\mathbb{D}^n$ are also presented.