A global approximation result by Bert Alan Taylor and the strong openness conjecture in \({\mathbb {C}}^n\) (Q1742927)

Let \(\varphi\) be a measurable function on \({\mathbb C}^n\), and let \(H(\varphi)\) be the space of entire functions \(f\) on \({\mathbb C}^n\) such that \[ \int_{{\mathbb C}^n}|f|^2e^{-\varphi}\,d\lambda<\infty\,, \] where \(\lambda\) is the Lebesgue measure. The authors prove the following global approximation theorem: If \(\{\varphi_j\}\) is an increasing sequence of plurisubharmonic functions on \({\mathbb C}^n\), \(\varepsilon>0\), \(\widetilde\varphi_j(z):=\varphi_j(z)+\varepsilon\log(1+\|z\|^2)\), and \(\widetilde\varphi=\lim_{j\to\infty}\widetilde\varphi_j\), then \(\bigcup_{j=1}^\infty H(\widetilde\varphi_j)\) is dense in \(H(\widetilde\varphi)\). This is an improvement of a result of \textit{B. A. Taylor} [Pac. J. Math. 36, 523--539 (1971; Zbl 0211.14904)]. This theorem leads to the following question, which is a natural version of the strong openness conjecture on \({\mathbb C}^n\): If \(\{\varphi_j\}\) is as above, \(\varphi=\lim_{j\to\infty}\varphi_j\), and \(f\) is an entire function with \(\int_{{\mathbb C}^n}|f|^2e^{-\varphi}\,d\lambda<\infty\), is it true that \(\int_{{\mathbb C}^n}|f|^2e^{-\varphi_k}\,d\lambda<\infty\) holds for all \(k\) sufficiently large? The question is answered in the negative by the authors.