Cohérence des faisceaux d'idéaux multiplicateurs avec estimations. (Coherence of multiplier ideal sheaves with estimates). (Q1426567)

Let \(\Omega\subset\mathbb C^n\) be a bounded pseudoconvex open set and let \(\phi\in\mathcal{PSH}(\Omega)\). For any positive integer \(m\), let \(\mathcal I(m\phi)\) be the multiplier ideal sheaf of germs of functions \(f\in\mathcal O(U)\), \(U\subset\Omega\), such that \(\int_U| f| ^2e^{-2m\phi}\,d\lambda<+\infty\) (\(\lambda\) denotes the Lebesgue measure). Let, moreover, \(\mathcal H_{\Omega}(m\phi):=\{f\in\mathcal O(\Omega): \int_\Omega| f| ^2e^{-2m\phi}\,d\lambda<+\infty\}\). \textit{A. M. Nadel} [Ann. Math. (2) 132, 549--596 (1990; Zbl 0731.53063)] proved that \(\mathcal I(m\phi)\) is a coherent sheaf generated by any orthonormal basis \((\sigma_j)_{j=0}^\infty\) of \(\mathcal H_{\Omega}(m\phi)\). The author presents the following version of this theorem with estimates. For every ball \(B(a,r)\subset\subset\Omega\) there exists a constant \(C(n)>0\) such that for any section \(f\in\Gamma(B(a,r),\mathcal I(m\phi))\) there exist functions \((b_j)_{j=0}^\infty\subset\mathcal O(B(a,r'))\) with \(r':=\frac{r}{\sqrt{nC(n)}}(\frac{r}{d})^{n+2}\), \(d:=\text{diam}\Omega\), such that \(f=\sum_{j=0}^\infty b_j\sigma_j\) on \(B(a,r')\) and \[ \sup_{B(a,r')}\sum_{j=0}^\infty| b_j| ^2\leq\frac1{(1-r/d)^2} \frac{C(n)}{(r/d)^{2(n+2)}}\int_{B(a,r)}| f| ^2e^{-2m\phi}\,d\lambda. \]