Plurisubharmonic functions with logarithmic singularities
Annales de l'Institut Fourier, Volume 38 (1988) no. 4, pp. 133-171.

To a plurisubharmonic function u on C n with logarithmic growth at infinity, we may associate the Robin function

ρ u ( z ) = lim sup λ u ( λ z ) - log ( λ z )

defined on P n-1 , the hyperplane at infinity. We study the classes L + , and (respectively) L p of plurisubharmonic functions which have the form u=log(1+|z|)+O(1) and (respectively) for which the function ρ u is not identically -. We obtain an integral formula which connects the Monge-Ampère measure on the space C n with the Robin function on P n-1 . As an application we obtain a criterion on the convergence of the Monge-Ampère measures of a sequence of functions in L + which is equivalent to the convergence of the associated Robin functions. As a consequence, it is shown that a polar set E is contained in {Ψ=-} for some ΨL ρ , and so the polar propagator E * , given as the intersection of the sets {Ψ=-} containing E, is polar. Ir A is an algebraic hypersurface which is disjoint from E, then E * cannot contain A.

On associe à une fonction u, plurisousharmonique dans C n de croissance logarithmique à l’infini, la fonction de Robin

ρ u ( z ) = lim sup λ u ( λ z ) - log ( λ z )

dans l’hyperplan P n-1 à l’infini. On étudie L + , la classe des fonctions de la forme u=log(1+|z|)+O(1) et L p , la classe des fonctions pour lesquelles la fonction ρ u n’est pas identiquement -. On obtient une formule intégrale qui relie la mesure de Monge-Ampère sur l’espace C n et la fonction de Robin. Sous titre d’application, on donne un critère sur les mesures de Monge-Ampère d’une suite de fonctions {u j }L + qui est nécessaire et suffisante pour la convergence des fonctions de Robin {r u }. Par conséquent, on trouve qu’un ensemble polaire E est contenu dans {Ψ=-} pour une fonction uL ρ , donc que l’ensemble de propagation E * , l’intersection des ensembles {Ψ=-} contenant E, est polaire. Soit A une hypersurface algébrique, EA=, alors E * ne contient pas A.

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     author = {Bedford, E. and Taylor, B. A.},
     title = {Plurisubharmonic functions with logarithmic singularities},
     journal = {Annales de l'Institut Fourier},
     pages = {133--171},
     publisher = {Institut Fourier},
     address = {Grenoble},
     volume = {38},
     number = {4},
     year = {1988},
     doi = {10.5802/aif.1152},
     mrnumber = {90f:32016},
     zbl = {0626.32022},
     language = {en},
     url = {http://archive.numdam.org/articles/10.5802/aif.1152/}
}
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Bedford, E.; Taylor, B. A. Plurisubharmonic functions with logarithmic singularities. Annales de l'Institut Fourier, Volume 38 (1988) no. 4, pp. 133-171. doi : 10.5802/aif.1152. http://archive.numdam.org/articles/10.5802/aif.1152/

[AT] H. Alexander and B. A. Taylor, Comparison of two capacities in ℂn, Math. Z., 186 (1984), 407-417. | MR | Zbl

[BT 1] E. Bedford and B. A. Taylor, A new capacity for plurisubharmonic functions, Acta Math., 149 (1982), 1-40. | MR | Zbl

[BT 2] E. Bedford and B. A. Taylor, Fine topology, Silov boundary, and (ddc)n, J. Funct. Anal., 72 (1987), 225-251. | MR | Zbl

[Car] L. Carleson, Selected Problems in Exceptional Sets, Van Nostrand Math. Studies # 13, D. Van Nostrand (1968). | Zbl

[Ceg] U. Cegrell, An estimate of the complex Monge-Ampere operator. Analytic functions. Proceedings, Blazejwko 1982. Lecture Notes in Math., 1039, pp. 84-87. Berlin, Heidelberg, New York, Springer, 1983. | Zbl

[D 1] J.-P. Demailly, Mesure de Monge-Ampère et caractérisation des variétés algébriques affines, mémoire (nouvelle série) n° 19, Soc. Math. de France, 1985. | Numdam | MR | Zbl

[D 2] J.-P. Demailly, Mesures de Monge-Ampère et mesures pluriharmoniques, Math. Z:, 194 (1987), 519-564. | MR | Zbl

[F] H. Federer, Geometric Measure Theory, Springer-Verlag, New York, 1969. | MR | Zbl

[J] B. Josefson, On the equivalence between locally polar and globally polar sets for plurisubharmonic functions on ℂn, Ark. Math., 16 (1978), 109-115. | MR | Zbl

[Km] M. Klimek, Extremal plurisubharmonic functions and invariant pseudo-distances, Bull. Soc. Math. France, 113 (1985), 123-142. | Numdam | MR | Zbl

[K 1] S. Kolodziej, Logarithmic capacity in ℂn, to appear in Ann. Pol. Math. | Zbl

[K 2] S. Kolodziej, On capacities associated to the Siciak extremal function (preprint). | Zbl

[Lp 1] L. Lempert, La métrique de Kobayashi et la représentation des domaines sur la boule, Bull. Soc. Math. France, 109 (1981), 427-474. | Numdam | MR | Zbl

[Lp 2] L. Lempert, Solving the degenerate Monge-Ampere equation with one concentrated singularity, Math. Ann., 263 (1983), 515-532. | MR | Zbl

[L] N. Levenberg, Capacities in several complex variables, Dissertation, The University of Michigan, 1984.

[Sa] A. Sadullaev, Plurisubharmonic measures and capacities on complex manifolds, Russian Math. Surveys, 36 (1981), 61-119. | MR | Zbl

[Si 1] J. Siciak, Extremal plurisubharmonic functions in ℂn, Ann. Polon. Math., 39 (1981). | MR | Zbl

[Si 2] J. Siciak, Extremal plurisubharmonic functions and capacities in ℂn, Sophia University, Tokyo, 1982. | Zbl

[Si 3] J. Siciak, On logarithmic capacities and pluripolar sets in ℂn (preprint).

[So] S. Souhail, Relations entre différentes notions d'ensembles pluripolaires complets dans ℂn, Thèse, L'Université Paul Sabatier de Toulouse, 1987.

[T] B. A. Taylor, An estimate for an extremal plurisubharmonic function on ℂn, Séminaire P. Lelong, P. Dolbeault, H. Skoda, 1982-1983, Lectures Notes in Math., 1028 (1983), 318-328. | MR | Zbl

[Z] V. P. Zaharjuta, Transfinite diameter Čebyšev constants, and capacity for compacta in ℂn, Math. USSR Sbornik, 25 (1975), 350-364. | Zbl

[Ze] A. Zeriahi, Ensembles pluripolaires exceptionnels pour croissance partielle des fonctions holomorphes, preprint. | Zbl

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