Theorem 5.37.15.5433D9F2E00C Let $\psi $ and $\phi _1,\ldots ,\phi _ r$ be plurisubharmonic functions on $\Omega $, with $r\geq 1$ and $\phi _ i<0$, and choose $p_ i>1$. For holomorphic $f$ with
\[ I=\| f\| _{\psi +\sum _ i\phi _ i}^2<+\infty , \]
put
\[ c=\inf \left\{ \| g\| _\psi ^2:g\in \mathcal O(\Omega ),\quad f-g\in \sum _{i=1}^ r\Gamma (\Omega ,\mathcal I(p_ i\phi _ i+\psi ))\right\} . \]
Then
646E9C7EF985
5.37.16
\begin{equation} c\leq \prod _{i=1}^ r\left(1-\frac1{p_ i}\right)I. \end{equation}
For $r\leq n$, this constant is optimal for the stated class of data.
Proof.
Set $\psi _ i=\psi +\sum _{j>i}\phi _ j$, $a_ i=1-1/p_ i$, and $f_0=f$. For arbitrary $\varepsilon _ i>0$, proposition 5.37.14 successively gives holomorphic $f_ i$ such that
\[ \begin{aligned} \| f_ i\| _{\psi _ i}^2& \leq a_ i\| f_{i-1}\| _{\psi _ i+\phi _ i}^2+\varepsilon _ i,\\ f_{i-1}-f_ i& \in \Gamma (\Omega ,\mathcal I(p_ i\phi _ i+\psi _ i)). \end{aligned} \]
Since $\psi _ i\leq \psi $, these ideals are contained in $\mathcal I(p_ i\phi _ i+\psi )$. As $f-f_ r$ is the sum of the differences $f_{i-1}-f_ i$, the function $f_ r$ is admissible for $c$. Moreover, $\psi _ i+\phi _ i=\psi _{i-1}$ for $i\geq 1$ and $\psi _ r=\psi $, so iteration gives
\[ c\leq \left(\prod _ i a_ i\right)I +\sum _{i=1}^ r\varepsilon _ i\prod _{j>i}a_ j. \]
Letting every $\varepsilon _ i$ tend to zero proves (5.37.16).
For sharpness when $r\leq n$, take the unit polydisk $\Omega =\Delta ^ n$, $f=1$, $\psi =0$, and $\phi _ i=p_ i^{-1}\log |z_ i|^2$ for $i\leq r$. By lemma 5.37.1, $\mathcal I(p_ i\phi _ i)=(z_ i)$. Every admissible $g$ equals one on $Y=\{ z_1=\cdots =z_ r=0\} $, hence $g(0)=1$. Orthogonality of the monomials on the polydisk gives $\| g\| _0^2\geq \pi ^ n$, with equality for $g=1$. Direct radial integration gives $I=\pi ^ n\prod _ i p_ i/(p_ i-1)$, so equality holds in (5.37.16).
$\square$
Pome
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