Pointwise estimate for the Bergman kernel of the weighted Bergman spaces with exponential type weights

Saïd Asserda; Amal Hichame

doi:10.1016/j.crma.2013.11.001

Complex analysis

Pointwise estimate for the Bergman kernel of the weighted Bergman spaces with exponential type weights
[Estimation ponctuelle du noyau de Bergman des espaces à poids de type exponentiel]

Présenté par : Jean-Pierre Demailly

Saïd Asserda ¹ ; Amal Hichame ²

¹ Ibn Tofail University, Faculty of Sciences, Department of Mathematics, PO 242 Kenitra, Morocco
² Regional Centre of Trades of Education and Training, Kenitra, Morocco

Comptes Rendus. Mathématique, Volume 352 (2014) no. 1, pp. 13-16.

Résumé
Abstract

Soit ${AL}_{ϕ}^{2} (D)$ le sous-espace fermé de $L^{2} (D, e^{- 2 ϕ} d λ)$ formé des fonctions holomorphes sur le disque unité $D$ . Pour une classe de fonctions sous-harmoniques $ϕ : D \to D$ , on établit une estimation ponctuelle du noyau de Bergman de ${AL}_{ϕ}^{2} (D)$ .

Let ${AL}_{ϕ}^{2} (D)$ denote the closed subspace of $L^{2} (D, e^{- 2 ϕ} d λ)$ consisting of holomorphic functions in the unit disc $D$ . For certain class of subharmonic functions $ϕ : D \to D$ , we prove an upper pointwise estimate for the Bergman kernel for ${AL}_{ϕ}^{2} (D)$ .

Reçu le : 2013-09-19
Accepté le : 2013-11-04
Publié le : 2013-12-17

DOI : 10.1016/j.crma.2013.11.001

Affiliations des auteurs :

Saïd Asserda ¹ ; Amal Hichame ²

¹ Ibn Tofail University, Faculty of Sciences, Department of Mathematics, PO 242 Kenitra, Morocco
² Regional Centre of Trades of Education and Training, Kenitra, Morocco

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     author = {Sa{\"\i}d Asserda and Amal Hichame},
     title = {Pointwise estimate for the {Bergman} kernel of the weighted {Bergman} spaces with exponential type weights},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {13--16},
     publisher = {Elsevier},
     volume = {352},
     number = {1},
     year = {2014},
     doi = {10.1016/j.crma.2013.11.001},
     language = {en},
}

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Saïd Asserda; Amal Hichame. Pointwise estimate for the Bergman kernel of the weighted Bergman spaces with exponential type weights. Comptes Rendus. Mathématique, Volume 352 (2014) no. 1, pp. 13-16. doi : 10.1016/j.crma.2013.11.001. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2013.11.001/

Bibliographie
Cité par

[1] H. Arroussi; J. Pau Reproducing kernel estimates, bounded projections and duality on large weighted Bergman spaces, Sep. 2013 | arXiv

[2] D. Azagra; J. Ferrera; F. Lopez-Mesas; Y. Rangel Smooth approximation of Lipschitz functions on Riemannian manifolds, J. Math. Anal. Appl., Volume 326 (2007), pp. 1370-1378

[3] B. Berndtsson Weighted estimates for the $\bar{\partial}$ -equation, Columbus, OH, 1999 (Ohio State Univ. Math. Res. Inst. Publ.), Volume vol. 9, De Gruyter, Berlin (2001), pp. 43-57

[4] M. Christ On the $\bar{\partial}$ -equation in weighted $L^{2}$ norm in $C$ , J. Geom. Anal., Volume 1 (1991) no. 3, pp. 193-230

[5] H. Delin Pointwise estimates for the weighted Bergman projection kernel in $C^{n}$ using a weighted $L^{2}$ estimate for the $\bar{\partial}$ equation, Ann. Inst. Fourier (Grenoble), Volume 48 (1998) no. 4, pp. 967-997

[6] R.E. Greene; H. Wu $C^{\infty}$ approximations of convex, subharmonic, and plurisubhrmonic functions, Ann. Sci. Ec. Norm. Super. (4), Volume 12 (1979) no. 1, pp. 47-84

[7] P. Lin; R. Rochberg Trace ideal criteria for Toeplitz and Hankel operators on the weighted Bergman spaces with exponential type weights, Pac. J. Math., Volume 173 (1996) no. 1, pp. 127-146

[8] N. Lindholm Sampling in weighted $L^{p}$ spaces of entire function in $C^{n}$ and estimates of the Bergman kernel, J. Funct. Anal., Volume 182 (2001), pp. 390-426

[9] J. Marzo; J. Ortega-Cerdà Pointwise estimates for the Bergman kernel of the weighted Fock space, J. Geom. Anal., Volume 19 (2009), pp. 890-910

[10] V.L. Oleinik; B.S. Pavlov Imbedding theorems for weighted classes of harmonic and analytic functions, J. Sov. Math., Volume 2 (1974), pp. 135-142 translation in: Zap. Nauchn. Sem. LOMI Steklov 22 (1971)

[11] V.L. Oleinik; G.S. Perelʼman Carlesonʼs Imbedding theorems for a weighted Bergman spaces, Math. Notes, Volume 7 (1990), pp. 577-581

[12] A.P. Schuster; D. Varolin New estimates for the minimal $L^{2}$ solution of $\bar{\partial}$ and application to geometric function theory in weighted Bergman spaces, J. Reine Angew. Math. (2013) | DOI

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