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Zero sets of holomorphic functions in the bidisk. (English) Zbl 0911.32003

A closed positive \((1,1)\)-current \(\Theta\) on the bidisk \(D\) in \(\mathbb{C}^2\) can be expressed in coordinates as \(\Theta(z)=i \sum^2_{i,j=1} \theta_{ij}(z)dz_i\wedge d\overline z_j\), where \(\theta_{11}(z_1,z_2)\) \((\theta_{22}(z_1,z_2))\) is a positive measure in the first (in the second) variable if we fix \(z_2\) (if we fix \(z_1)\). Therefore, the author defines \(\theta_{11} (D_{z_1},z_2)=\int_{\zeta\in D_{z_1}}d\theta_{11}(\zeta,z_2)\), where \(D_z=\left \{\zeta\in D: {|\zeta-z |\over| 1-\overline \zeta z|}< \varepsilon\right\}\). Now we can state the main result of the article under review.
Theorem 2.1. The equation \(i\partial\overline\partial u= \Theta\) has a solution \(u\in L^p(D)\), \(1\leq p<\infty\), iff the function \(f(z_1, z_2)= \theta_{11}(D_{z_1},z_2)+\theta_{22}(z_1,D_{z_2})\) belongs to \(L^p(D)\).
Let \(n_V(z)\) be the number of points (counted with multiplicity) that the analytic variety \(V\) meets the cross \(C_z=\{\zeta\in D:\zeta_1=z_1,\zeta_2\in D_{z_2}\}\cup\{\zeta\in D:\zeta_2=z_2,\zeta_1\in D_{z_1}\}\).
Corollary 2.3. The analytic variety \(V\) is the zero set of a function \(f\) with \(\log| f|\in L^p(D)\) iff \(n_V\in L^p(D)\).

MSC:

32A35 \(H^p\)-spaces, Nevanlinna spaces of functions in several complex variables
32A22 Nevanlinna theory; growth estimates; other inequalities of several complex variables
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