Subharmonicity properties of the Bergman kernel and some other functions associated to pseudoconvex domains
Annales de l'Institut Fourier, Volume 56 (2006) no. 6, pp. 1633-1662.

Let D be a pseudoconvex domain in t k × z n and let φ be a plurisubharmonic function in D. For each t we consider the n-dimensional slice of D, D t ={z;(t,z)D}, let φ t be the restriction of φ to D t and denote by K t (z,ζ) the Bergman kernel of D t with the weight function φ t . Generalizing a recent result of Maitani and Yamaguchi (corresponding to n=1 and φ=0) we prove that logK t (z,z) is a plurisubharmonic function in D. We also generalize an earlier results of Yamaguchi concerning the Robin function and discuss similar results in the setting of  n .

Soit D un domaine pseudoconvexe en t k × z n et soit φ une fonction plurisousharmonique dans D. Pour t fixé, soit D t ={z;(t,z)D} la tranche correspondante de D, φ t la restriction de φ à D t , et K t (z,ζ) le noyau de Bergman pour le domaine D t et le poid φ t . En généralisant un résultat récent de Maitani et Yamaguchi (correspondant à n=1 et φ=0), on montre que logK t (z,z) est plurisousharmonique en D. On donne aussi une généralisation d’un résultat de Yamaguchi concernant la fonction de Robin et on discute des résultats du même style pour  n .

DOI: 10.5802/aif.2223
Classification: 32A25
Keywords: Bergman spaces, plurisubharmonic function, ¯-equation, Lelong number
Berndtsson, Bo 1

1 Chalmers University of Technology and the University of Göteborg Department of Mathematics 412 96 Göteborg (Sweden)
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Berndtsson, Bo. Subharmonicity properties of the Bergman kernel and some other functions associated to pseudoconvex domains. Annales de l'Institut Fourier, Volume 56 (2006) no. 6, pp. 1633-1662. doi : 10.5802/aif.2223. https://aif.centre-mersenne.org/articles/10.5802/aif.2223/

[1] Ball, K.; Barthe, F.; Naor, A. Entropy jumps in the presence of a spectral gap, Duke Math. J., Volume 119 (2003), pp. 41-63 | DOI | MR | Zbl

[2] Berndtsson, B. Prekopa’s theorem and Kiselman’s minimum principle for plurisubharmonic functions, Math. Ann., Volume 312 (1998), pp. 785-792 | DOI | MR | Zbl

[3] Brascamp, H. J.; Lieb, E. H. On extensions of the Brunn-Minkowski and Prekopa-Leindler theorems, including inequalities for log concave functions, and with an application to the diffusion equation, J. Funct. Anal., Volume 22 (1976), pp. 366-389 | DOI | MR | Zbl

[4] Bruna, J.; Burgués, J. Holomorphic approximation and estimates for the ¯-equation on strictly pseudoconvex nonsmooth domains, Duke Math. J., Volume 55 (1987), pp. 539-596 | DOI | MR | Zbl

[5] Cardaliaguet, P.; Tahraoui, R. On the strict concavity of the harmonic radius in dimension N3, J. Math. Pures Appl. (9), Volume 81 (2002), pp. 223-240 | MR | Zbl

[6] Cordero-Erausquin, D. Santaló’s inequality on n by complex interpolation, C. R. Math. Acad. Sci. Paris, Volume 334 (2002), pp. 767-772 | MR | Zbl

[7] Cordero-Erausquin, D. On Berndtsson’s generalization of Prekopa’s theorem, Math. Z., Volume 249 (2005), pp. 401-410 | DOI | MR | Zbl

[8] Demailly, J.-P. Estimations L 2 pour l’opérateur ¯ d’un fibré vectoriel holomorphe semi-positif au-dessus d’une variété kählérienne complète, Ann. Sci. École Norm. Sup. (4), Volume 15 (1982), pp. 457-511 | Numdam | MR | Zbl

[9] Hörmander, L. L 2 -estimates and existence theorems for the ¯-operator, Acta Math., Volume 113 (1965), pp. 89-152 | DOI | MR | Zbl

[10] Kiselman, C. O. The partial Legendre transformation for plurisubharmonic functions, Invent. Math., Volume 49 (1978), pp. 137-148 | DOI | MR | Zbl

[11] Kiselman, C. O. Densité des fonctions plurisousharmoniques, Bull. Soc. Math. France., Volume 107 (1979), pp. 295-304 | Numdam | MR | Zbl

[12] Kiselman, C. O. Attenuating the singularities of plurisubharmonic functions, Ann. Polon. Math., Volume 60 (1994), pp. 173-197 | MR | Zbl

[13] Levenberg, N.; Yamaguchi, H. Robin functions for complex manifolds and applications (2004) (Manuscript)

[14] Maitani, F.; Yamaguchi, H. Variation of Bergman metrics on Riemann surfaces, Math. Annal., Volume 330 (2004), pp. 477-489 | DOI | MR | Zbl

[15] Prekopa, A. On logarithmic concave measures and functions, Acad. Sci. Math. (Szeged), Volume 34 (1973), pp. 335-343 | MR | Zbl

[16] Siu, Y.-T. Analyticity of sets associated to Lelong numbers and the extension of closed positive currents, Invent. Math., Volume 27 (1974), pp. 53-156 | DOI | MR | Zbl

[17] Skoda, H. Sous-ensembles analytiques d’ordre fini ou infini dans  n , Bull. Soc. Math. France, Volume 100 (1972), pp. 353-408 | Numdam | MR | Zbl

[18] Yamaguchi, H. Variations of pseudoconvex domains over  n , Michigan Math. J., Volume 36 (1989), pp. 415-457 | DOI | MR | Zbl

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