Equivalent Bergman spaces with inequivalent weights (Q1725859)

Let $D\subset\mathbb C^N$ be a domain and let $\mu$ be a weight, i.e., a positive measurable function on $D$. Let $L^2(D,\mu)$ be the Hilbert space of measurable functions with inner product \[ \langle f,g\rangle_{\mu}:=\int_Df(z)\overline{g(z)}\mu(z)\,dA(z), \] and let $L^2_H(D,\mu)$ denote the subspace of $L^2(D,\mu)$ consisting of those functions that are holomorphic. A weight $\mu$ is called admissible, if for each $z\in D$ the evaluation functional $f\longmapsto f(z)$ is continuous on $L^2_H(D,\mu)$ and if $L^2_H(D,\mu)$ is a closed subspace of $L^2(D,\mu)$. By $K_{D,\mu}$ we denote the weighted Bergman kernel of $D$ (with respect to the admissible weight $\mu$). Finally, we say that two admissible weigths $\mu_1$ and $\mu_2$ are equivalent, if $L^2_H(D,\mu_1)=L^2_H(D,\mu_2)$ as sets. \par The main results of the paper under review are the following two theorems, which answer affirmatively two questions posed by \textit{A. Perälä} in [J. Geom. Anal. 28, No. 2, 1716--1727 (2018; Zbl 1404.30064)]. \begin{itemize} \item[1.] If $\mu$ is an admissible weight on a domain $D\subset\mathbb C^N$, then there exists an admissible weight $\mu^*$ equivalent to $\mu$ such that $K_{D,\mu^*}$ has zeroes. \item[2.] If \[ W(z):=\frac{1}{2\pi}|z|^n\exp\left(-\alpha|z|^{2m}\right),\quad n>-2,\ \alpha,m>0,\ m\notin\mathbb Z, \] then $K_{\mathbb C,W}(\cdot,w)$ has infinitely many zeroes for any $w\in\mathbb C\setminus\{0\}$. \end{itemize}