no. 4
Pointwise estimates for the weighted Bergman projection kernel in ℂ n , using a weighted L 2 estimate for the ∂ ¯ equation
Annales de l'Institut Fourier, Tome 48 (1998) no. 4, pp. 967-997.

Nous obtenons des estimations L 2 à poids pour la solution canonique de l’équation ∂ ¯ dans L 2 (ℂ n ,e -ϕ dλ), où Ω est un domaine pseudoconvexe et ϕ une fonction strictement plurisousharmonique. Ces estimations sont ensuite utilisées pour démontrer des estimations ponctuelles pour le noyau du projecteur de Bergman dans L 2 (ℂ n ,e -ϕ dλ). Le poids est utilisé pour obtenir un facteur e -ϵρ(z,ζ) dans l’estimation du noyau, où ρ est la distance associée à la métrique kählérienne définie par i∂∂ ¯ϕ.

Weighted L 2  estimates are obtained for the canonical solution to the ∂ ¯ equation in L 2 (ℂ n ,e -ϕ dλ), where Ω is a pseudoconvex domain, and ϕ is a strictly plurisubharmonic function. These estimates are then used to prove pointwise estimates for the Bergman projection kernel in L 2 (ℂ n ,e -ϕ dλ). The weight is used to obtain a factor e -ϵρ(z,ζ) in the estimate of the kernel, where ρ is the distance function in the Kähler metric given by the metric form i∂∂ ¯ϕ.

@article{AIF_1998__48_4_967_0,
     author = {Delin, Henrik},
     title = {Pointwise estimates for the weighted {Bergman} projection kernel in ${\mathbb {C}}^n$, using a weighted $L^2$ estimate for the $\bar{\partial }$ equation},
     journal = {Annales de l'Institut Fourier},
     pages = {967--997},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {48},
     number = {4},
     year = {1998},
     doi = {10.5802/aif.1645},
     zbl = {0918.32007},
     mrnumber = {99j:32027},
     language = {en},
     url = {https://aif.centre-mersenne.org/articles/10.5802/aif.1645/}
}
TY  - JOUR
AU  - Delin, Henrik
TI  - Pointwise estimates for the weighted Bergman projection kernel in ${\mathbb {C}}^n$, using a weighted $L^2$ estimate for the $\bar{\partial }$ equation
JO  - Annales de l'Institut Fourier
PY  - 1998
SP  - 967
EP  - 997
VL  - 48
IS  - 4
PB  - Association des Annales de l’institut Fourier
UR  - https://aif.centre-mersenne.org/articles/10.5802/aif.1645/
DO  - 10.5802/aif.1645
LA  - en
ID  - AIF_1998__48_4_967_0
ER  - 
%0 Journal Article
%A Delin, Henrik
%T Pointwise estimates for the weighted Bergman projection kernel in ${\mathbb {C}}^n$, using a weighted $L^2$ estimate for the $\bar{\partial }$ equation
%J Annales de l'Institut Fourier
%D 1998
%P 967-997
%V 48
%N 4
%I Association des Annales de l’institut Fourier
%U https://aif.centre-mersenne.org/articles/10.5802/aif.1645/
%R 10.5802/aif.1645
%G en
%F AIF_1998__48_4_967_0
Delin, Henrik. Pointwise estimates for the weighted Bergman projection kernel in ${\mathbb {C}}^n$, using a weighted $L^2$ estimate for the $\bar{\partial }$ equation. Annales de l'Institut Fourier, Tome 48 (1998) no. 4, pp. 967-997. doi : 10.5802/aif.1645. https://aif.centre-mersenne.org/articles/10.5802/aif.1645/

[1] S. Bergman, The kernel function and conformal mapping, American Mathematical Society, Providence, R.I., revised ed., 1970, Mathematical Surveys, no V. | MR | Zbl

[2] B. Berndtsson, Uniform estimates with weights for the ∂-equation, to appear in J. Geom. Analysis. | Zbl

[3] I. Chavel, Riemannian geometry - a modern introduction, vol. 108 of Cambridge Tracts in Mathematics, Cambridge University Press, Cambridge, 1993. | MR | Zbl

[4] M. Christ, On the ∂ equation in weighted L2 norms in ℂ1, J. Geom. Anal., 1 (1991), 193-230. | MR | Zbl

[5] K. Diederich and G. Herbort, Extension of holomorphic L2-functions with weighted growth conditions, Nagoya Math. J., 126 (1992), 141-157. | MR | Zbl

[6] K. Diederich and T. Ohsawa, An estimate for the Bergman distance on pseudoconvex domains, Ann. of Math. (2), 141 (1995), 181-190. | MR | Zbl

[7] H. Donnelly and C. Fefferman, L2-cohomology and index theorem for the Bergman metric, Ann. of Math. (2), 118 (1983), 593-618. | MR | Zbl

[8] S. Dragomir, On weighted Bergman kernels of bounded domains, Studia Math., 108 (1994), 149-157. | MR | Zbl

[9] G. M. Henkin and J. Leiterer, Theory of functions on complex manifolds, vol. 79 of Monographs in Mathematics, Birkhäuser Verlag, Basel, 1984. | MR | Zbl

[10] L. Hörmander, L2 estimates and existence for the ∂ operator, Acta Mathematica, 113 (1965), 89-152. | Zbl

[11] N. Kerzman, The Bergman kernel function. Differentiability at the boundary, Math. Ann., 195 (1972), 149-158.

[12] P. Lelong and L. Gruman, Entire functions of several complex variables, Grundlehren der Mathematischen Wissenschaften [Fundamental Principales of Mathematical Sciences], 282, Springer-Verlag, Berlin, 1986. | MR | Zbl

[13] J. D. Mcneal, Boundary behavior of the Bergman kernel function in ℂ2, Duke Math. J., 58 (1989), 499-512. | MR | Zbl

[14] J. D. Mcneal, On large values of L2 holomorphic functions, Math. Res. Lett., 3 (1996), 247-259. | MR | Zbl

[15] A. Nagel, J.-P. Rosay, E. M. Stein, and S. Wainger, Estimates for the Bergman and Szegö kernels in ℂ2, Ann. of Math. (2), 129 (1989), 113-149. | MR | Zbl

[16] T. Ohsawa and K. Takegoshi, On the extension of L2 holomorphic functions, Math. Z., 195 (1987), 197-204. | MR | Zbl

[17] R. Schoen and S.-T. Yau, Lectures on differential geometry, Conference Proceedings and Lecture Notes in Geometry and Topology, I, International Press, Cambridge, MA, 1994. | MR | Zbl

[18] Y.-T. Siu, Complex-analyticity of harmonic maps, vanishing and Lefschetz theorems, J. Differential Geometry, 17 (1982), 55-138. | MR | Zbl

[19] Y.-T. Siu, The Fujita conjecture and the extension theorem of Ohsawa-Takegoshi, in Geometric Complex Analysis (Hayma, 1995), World Sci. Publishing, River Edge, NJ (1996), 577-592. | MR | Zbl

[20] R. O. Wells, Jr, Differential analysis on complex manifolds, Prentice-Hall, 1973. | Zbl

Cité par Sources :