A Hilbert Lemniscate Theorem in $\mathbb{C}^2$

Bloom, Thomas; Levenberg, Norman; Lyubarskii, Yu.

doi:10.5802/aif.2411

A Hilbert Lemniscate Theorem in

ℂ^{2}

[Un théorème de la lemniscate de Hilbert dans

ℂ^{2}

]

Bloom, Thomas ¹ ; Levenberg, Norman ² ; Lyubarskii, Yu.³

¹ University of Toronto Toronto (Canada)
² Indiana University Bloomington, IN 47405 (USA)
³ Norwegian University of Science and Technology Trondheim, 7491 (Norway)

Annales de l'Institut Fourier, Tome 58 (2008) no. 6, pp. 2191-2220.

Résumé
Abstract

Pour un compact $K$ dans $C^{2}$ , regulier, pôlynomiallement convexe et cerclé, on construit une suite de paires ${P_{n}, Q_{n}}$ avec $P_{n}, Q_{n}$ pôlynomes homogènes en deux variables et $\deg P_{n} =$ $\deg Q_{n} = n$ tel que les ensembles $K_{n} : = {(z, w) \in C^{2} : | P_{n} (z, w) | \leq 1, | Q_{n} (z, w) | \leq 1}$ font une approximation de $K$ et quand $K$ est la fermeture d’un domaine strictement pseudoconvexe les mesures de comptage normalisées associées à l’ensemble fini ${P_{n} = Q_{n} = 1}$ tendent vers la mesure de Monge-Ampère pour $K$ . L’élément principal est un théorème d’approximation pour les fonctions sousharmoniques de croissance logarithmique à une variable.

For a regular, compact, polynomially convex circled set $K$ in $C^{2}$ , we construct a sequence of pairs ${P_{n}, Q_{n}}$ of homogeneous polynomials in two variables with $\deg P_{n} =$ $\deg Q_{n} = n$ such that the sets $K_{n} : = {(z, w) \in C^{2} : | P_{n} (z, w) | \leq 1, | Q_{n} (z, w) | \leq 1}$ approximate $K$ and if $K$ is the closure of a strictly pseudoconvex domain the normalized counting measures associated to the finite set ${P_{n} = Q_{n} = 1}$ converge to the pluripotential-theoretic Monge-Ampère measure for $K$ . The key ingredient is an approximation theorem for subharmonic functions of logarithmic growth in one complex variable.

MR Zbl

DOI : 10.5802/aif.2411

Classification : 32U05, 32W20
Keywords: Logarithmic potential, Monge-Ampère measure, subharmonic functions, atomization
Mot clés : potentiel logarithmique, mesure de Monge-Ampère, fonctions sousharmoniques, atomisation

Affiliations des auteurs :

Bloom, Thomas ¹ ; Levenberg, Norman ² ; Lyubarskii, Yu. ³

¹ University of Toronto Toronto (Canada)
² Indiana University Bloomington, IN 47405 (USA)
³ Norwegian University of Science and Technology Trondheim, 7491 (Norway)

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     author = {Bloom, Thomas and Levenberg, Norman and Lyubarskii, Yu.},
     title = {A {Hilbert} {Lemniscate} {Theorem} in $\mathbb{C}^2$},
     journal = {Annales de l'Institut Fourier},
     pages = {2191--2220},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {58},
     number = {6},
     year = {2008},
     doi = {10.5802/aif.2411},
     zbl = {1152.32015},
     mrnumber = {2473634},
     language = {en},
     url = {http://www.numdam.org/articles/10.5802/aif.2411/}
}

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Bloom, Thomas; Levenberg, Norman; Lyubarskii, Yu. A Hilbert Lemniscate Theorem in $\mathbb{C}^2$. Annales de l'Institut Fourier, Tome 58 (2008) no. 6, pp. 2191-2220. doi : 10.5802/aif.2411. http://www.numdam.org/articles/10.5802/aif.2411/

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