no. 6
Subharmonicity properties of the Bergman kernel and some other functions associated to pseudoconvex domains
Berndtsson, Bo1
1Chalmers University of Technology and the University of Göteborg Department of Mathematics 412 96 Göteborg (Sweden)
Annales de l'Institut Fourier, Volume 56 (2006) no. 6, pp. 1633-1662

Let D be a pseudoconvex domain in ℂ t k ×ℂ 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 ℝ n .

Soit D un domaine pseudoconvexe en ℂ t k ×ℂ 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 ℝ n .

DOI: 10.5802/aif.2223
Classification: 32A25
Keywords: Bergman spaces, plurisubharmonic function, $\bar{\partial }$-equation, Lelong number
Mots-clés : espace de Bergman, fonction plurisousharmonique, équation $\bar{\partial }$, nombre de Lelong

Berndtsson, Bo  1

1 Chalmers University of Technology and the University of Göteborg Department of Mathematics 412 96 Göteborg (Sweden) 
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
@article{AIF_2006__56_6_1633_0,
     author = {Berndtsson, Bo},
     title = {Subharmonicity properties of the {Bergman} kernel and some other functions associated to pseudoconvex~domains},
     journal = {Annales de l'Institut Fourier},
     pages = {1633--1662},
     year = {2006},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {56},
     number = {6},
     doi = {10.5802/aif.2223},
     mrnumber = {2282671},
     zbl = {1120.32021},
     language = {en},
     url = {https://aif.centre-mersenne.org/articles/10.5802/aif.2223/}
}
TY  - JOUR
AU  - Berndtsson, Bo
TI  - Subharmonicity properties of the Bergman kernel and some other functions associated to pseudoconvex domains
JO  - Annales de l'Institut Fourier
PY  - 2006
SP  - 1633
EP  - 1662
VL  - 56
IS  - 6
PB  - Association des Annales de l’institut Fourier
UR  - https://aif.centre-mersenne.org/articles/10.5802/aif.2223/
DO  - 10.5802/aif.2223
LA  - en
ID  - AIF_2006__56_6_1633_0
ER  - 
%0 Journal Article
%A Berndtsson, Bo
%T Subharmonicity properties of the Bergman kernel and some other functions associated to pseudoconvex domains
%J Annales de l'Institut Fourier
%D 2006
%P 1633-1662
%V 56
%N 6
%I Association des Annales de l’institut Fourier
%U https://aif.centre-mersenne.org/articles/10.5802/aif.2223/
%R 10.5802/aif.2223
%G en
%F AIF_2006__56_6_1633_0

[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 | Zbl | MR

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

[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 | Zbl | MR

[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 | Zbl | MR

[5] Cardaliaguet, P.; Tahraoui, R. On the strict concavity of the harmonic radius in dimension N≥3, J. Math. Pures Appl. (9), Volume 81 (2002), pp. 223-240 | Zbl | MR

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

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

[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 | Zbl | MR | Numdam

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

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

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

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

[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 | Zbl | MR

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

[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 | Zbl | MR

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

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

Cited by Sources: