Subharmonicity properties of the Bergman kernel and some other functions associated to pseudoconvex domains
Annales de l'Institut Fourier, Volume 56 (2006) no. 6, p. 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 : https://doi.org/10.5802/aif.2223
Classification:  32A25
Keywords: Bergman spaces, plurisubharmonic function, ¯-equation, Lelong number
@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},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {56},
     number = {6},
     year = {2006},
     pages = {1633-1662},
     doi = {10.5802/aif.2223},
     mrnumber = {2282671},
     zbl = {1120.32021},
     language = {en},
     url = {http://www.numdam.org/item/AIF_2006__56_6_1633_0}
}
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. http://www.numdam.org/item/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., Tome 119 (2003), pp. 41-63 | Article | MR 1991646 | Zbl 1036.94003

[2] Berndtsson, B. Prekopa’s theorem and Kiselman’s minimum principle for plurisubharmonic functions, Math. Ann., Tome 312 (1998), pp. 785-792 | Article | MR 1660227 | Zbl 0938.32021

[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., Tome 22 (1976), pp. 366-389 | Article | MR 450480 | Zbl 0334.26009

[4] Bruna, J.; Burgués, J. Holomorphic approximation and estimates for the ¯-equation on strictly pseudoconvex nonsmooth domains, Duke Math. J., Tome 55 (1987), pp. 539-596 | Article | MR 904941 | Zbl 0645.32009

[5] Cardaliaguet, P.; Tahraoui, R. On the strict concavity of the harmonic radius in dimension N3, J. Math. Pures Appl. (9), Tome 81 (2002), pp. 223-240 | MR 1894062 | Zbl 1027.31003

[6] Cordero-Erausquin, D. Santaló’s inequality on n by complex interpolation, C. R. Math. Acad. Sci. Paris, Tome 334 (2002), pp. 767-772 | MR 1905037 | Zbl 1002.31003

[7] Cordero-Erausquin, D. On Berndtsson’s generalization of Prekopa’s theorem, Math. Z., Tome 249 (2005), pp. 401-410 | Article | MR 2115450 | Zbl 1079.32020

[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), Tome 15 (1982), pp. 457-511 | Numdam | MR 690650 | Zbl 0507.32021

[9] Hörmander, L. L 2 -estimates and existence theorems for the ¯-operator, Acta Math., Tome 113 (1965), pp. 89-152 | Article | MR 179443 | Zbl 0158.11002

[10] Kiselman, C. O. The partial Legendre transformation for plurisubharmonic functions, Invent. Math., Tome 49 (1978), pp. 137-148 | Article | MR 511187 | Zbl 0378.32010

[11] Kiselman, C. O. Densité des fonctions plurisousharmoniques, Bull. Soc. Math. France., Tome 107 (1979), pp. 295-304 | Numdam | MR 544525 | Zbl 0416.32007

[12] Kiselman, C. O. Attenuating the singularities of plurisubharmonic functions, Ann. Polon. Math., Tome 60 (1994), pp. 173-197 | MR 1301603 | Zbl 0827.32016

[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., Tome 330 (2004), pp. 477-489 | Article | MR 2099190 | Zbl 1077.32006

[15] Prekopa, A. On logarithmic concave measures and functions, Acad. Sci. Math. (Szeged), Tome 34 (1973), pp. 335-343 | MR 404557 | Zbl 0264.90038

[16] Siu, Y.-T. Analyticity of sets associated to Lelong numbers and the extension of closed positive currents, Invent. Math., Tome 27 (1974), pp. 53-156 | Article | MR 352516 | Zbl 0289.32003

[17] Skoda, H. Sous-ensembles analytiques d’ordre fini ou infini dans  n , Bull. Soc. Math. France, Tome 100 (1972), pp. 353-408 | Numdam | MR 352517 | Zbl 0246.32009

[18] Yamaguchi, H. Variations of pseudoconvex domains over  n , Michigan Math. J., Tome 36 (1989), pp. 415-457 | Article | MR 1027077 | Zbl 0692.31004