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

Keywords:

holomorphic functions; zero sets; bidisk
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References:

 [1] [AČ]Ahern, P. andČučković Ž., An invariant submean value property and hyponormal Toepliz operators, inHarmonic Analysis and Operator Theory (Marcantogini, S. A. M., Mendoza, G. A., Morán, M. D., Octavio, A. and Urbina, W. O., eds.), Contemp. Math.189, pp. 17–24. Amer. Math. Soc., Providence, R. I., 1995. · Zbl 0847.31003 [2] [A]Andersson, M., Solution formulas for the $$\partial \bar \partial$$ and weighted Nevanlinna classes in the polydisk,Bull. Soc. Math. France 109 (1985), 135–154. · Zbl 0598.32007 [3] [B]Beller, E., Zeros ofA p functions and related classes of analytic functions,Israel J. Math. 22 (1975), 68–80. · Zbl 0322.30028 [4] [BO]Bruna, J. andOrtega-Cerdà, J., OnL p -solutions of the Laplace equation and zeros of holomorphic functions, to appear inAnn. Scuola Norm. Sup. Pisa CL Sci. [5] [C]Charpentier, P., Caractérisations des zéros des fonctions des certaines classes de type Nevanlinna dans le bidisque,Ann. Inst. Fourier (Grenoble) 34:1 (1984), 57–98. · Zbl 0564.32002 [6] [K]Khenkin, G., Solutions with estimates of the H. Lewy and Poincaré-Lelong equations. The construction of functions of a Nevanlinna class with given zeros in a strictly pseudoconvex domain,Dokl. Akad. Nauk SSSR 224 (1975), 771–774. (Russian). English transl.:Soviet Math. Dokl. 16 (1975), 1310–1314. [7] [Le]Lelong, P.,Fonctionelles analytiques et fonctions entières (n variables), Les presses de l’Université de Montréal, Montréal, 1968. [8] [L1]Luecking, D., Multipliers of Bergman spaces into Lebesgue spaces,Proc. Edinburgh Math. Soc. 29 (1986), 125–131. · Zbl 0587.30048 [9] [L2]Luecking, D., Zero sequences for Bergman spaces,Complex Varialex Theory Appl. 30 (1996), 345–362. · Zbl 0871.30004 [10] [P]Pascuas, D.,Zeros i interpolació en espais de funcions, holomorfes del disc unitat, Ph. D. Thesis, Universitat Autònoma de Barcelona, 1988. [11] [R]Rudin, W.,Function Theory in the Unit Ball of C n , Grundlehren Math. Wiss.241, Springer-Verlag, Berlin-New York, 1980. · Zbl 0495.32001 [12] [S]Skoda, H., Valeurs au bord pour les solutions del l’operateurd”, et caractérisation des zéros des fonctions de la classe de Nevanlinna.Bull. Soc. Math. France 104 (1976), 225–299. · Zbl 0351.31007
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