In this subsection, $n\geq 2$, $B\subset \mathbb C^ n$ is the unit ball, and $\mathrm{d}V$ is Euclidean volume. If the weights have logarithmic poles along several hyperplanes through the origin, every admissible remainder must take a prescribed value at their common zero. On the other hand, the input norm stays finite as long as the combined radial exponent is below the dimension. Together these observations give a positive distance even in part of the range where the planar distance of theorem 5.37.22 vanishes.
Example 5.37.25.9F227CC9F378 Fix $r>n$ and $p>r/n$. There are negative plurisubharmonic functions $\phi _1,\ldots ,\phi _ r$ on $B$ for which
\[ I=\int _ B\mathrm e^{-\sum _ i\phi _ i}\, \mathrm{d}V<+\infty \]
and
4F7B8ACF0243
5.37.26
\begin{equation} \inf \left\{ \int _ B|g|^2\, \mathrm{d}V:g\in \mathcal O(B),\quad 1-g\in \sum _{i=1}^ r\Gamma (B,\mathcal I(p\phi _ i))\right\} =\frac{\pi ^ n}{n!}>0. \end{equation}
In particular, for $r/n<p\leq r$, the distance is positive although $\sum _ i1/p=r/p\geq 1$.
Proof.
Choose distinct real numbers $t_1,\ldots ,t_ r$ and unit linear forms
\[ \ell _ i(z)=\frac{\sum _{k=0}^{n-1}t_ i^ kz_{k+1}}{(\sum _{k=0}^{n-1}t_ i^{2k})^{1/2}}. \]
Any $n$ of these forms are linearly independent, since the determinant of their unnormalized coefficients is a nonzero Vandermonde determinant. Put $\phi _ i=p^{-1}\log |\ell _ i|^2$. Since the coefficient vectors of the $\ell _ i$ have norm one, $|\ell _ i(z)|<1$ on $B$, so the $\phi _ i$ are negative plurisubharmonic functions. By lemma 5.37.1, $\mathcal I(p\phi _ i)=(\ell _ i)$.
At a nonzero point, at most $n-1$ of the forms vanish. These forms are linearly independent and can be used as part of a holomorphic coordinate system, while the other forms are bounded away from zero on a small neighborhood of the point. Hence
\[ W(z)=\prod _{i=1}^ r|\ell _ i(z)|^{-2/p} \]
is locally integrable off the origin, because each factor $|\ell _ i|^{-2/p}$ has exponent $2/p<2$ (note that $p>r/n>1$). By compactness, $W$ is integrable on every closed annulus. Set $s=r/p$ and let $\mathrm{d}\sigma $ denote surface measure on the unit sphere. Homogeneity and polar integration over such an annulus show that
\[ A=\int _{|\zeta |=1}\prod _ i|\ell _ i(\zeta )|^{-2/p}\, \mathrm{d}\sigma (\zeta )<+\infty . \]
Tonelli then gives
\[ I=A\int _0^1\rho ^{2n-1-2s}\, \mathrm{d}\rho =\frac{A}{2(n-s)}<+\infty , \]
since $s<n$. This takes care of the common zero at the origin as well as of the individual hyperplanes.
Every global section of $(\ell _ i)$ vanishes at the origin. Thus every admissible $g$ in (5.37.26) satisfies $g(0)=1$. For each $z\in B$, the holomorphic function $\lambda \mapsto g(\lambda z)$ is defined on a disk of radius greater than one, and the submean inequality for the subharmonic function $|g(\lambda z)|^2$ gives
\[ \frac1{2\pi }\int _0^{2\pi }|g(\mathrm e^{\mathrm i\tau }z)|^2\, \mathrm{d}\tau \geq |g(0)|^2=1. \]
Integrating in $z$ and using the rotation invariance of $B$ proves $\int _ B|g|^2\, \mathrm{d}V\geq \operatorname {Vol}(B)=\pi ^ n/n!$, also when the integral is infinite. The constant $g=1$ is admissible and attains this bound.
$\square$
Hence, for these equal exponents, the optimal universal constant in an estimate of the distance by a multiple of $I$ is at least $\pi ^ n/(n!I)>0$. In particular, the vanishing of the optimal constant for $s\geq 1$ in theorem 5.37.22 does not extend to higher dimensions. The example does not determine the optimal constant for $r>n$.
Pome
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