Abstract
Definition of the Bergman space for an arbitrary operator is given. Sufficient conditions for the existence of the Bergman kernel for this space are obtained. For an elliptic operator, the Bergman kernel is represented via the Green function. Bibliography: 12 titles.
Similar content being viewed by others
Literature Cited
S. Bergman, “The kernel function and conformal mapping”,Math. Surveys, New York (1950).
N. Aronszajn, “Theory of reproducing kernels”,Trans. Am. Math. Soc.,68, 337–404 (1950).
N. N. Tarkhanov,The Parametrix Method in the Theory of Differential Complexes [in Russian], Novgorod (1990).
P. J. H. Hedenmalm “A computation of Green functions for the weighted biharmonic operators Δ|z|−2αΔ, with α>−1”,U.U.D.M. Report (1992).
P. J. H. Hedenmalm, “Open problems in the function theory of the Bergman space”,U.U.D.M. Report (1992).
P. R. Garabedian, “A partial differential equation arising in conformal mapping”,Pacific J. Math.,1, 485–524 (1951).
J. Hadamard,Mémoire sur le Problème d'Analyse Relatif à l'Équilibre des Plaques Élastiques Encastrées, Paris (1908).
B. V. Shabat,Introduction to Complex Analysis. Part II [in Russian], Moscow (1976).
L. Hörmander,Analysis of Linear Differential Operators with Partial Derivatives [Russian Translation], Mir, Moscow (1986).
V. A. Malyshev, “Hadamard conjecture and estimates of the Green function”,Algebra Analiz,4, 1–44 (1992).
V. A. Malyshev,Elliptic Differential Inequalities [in Russian], Moscow (1994).
I. N. Vekua,New Methods of Solution of Elliptic Equations [in Russian], Moscow (1948).
Additional information
Dedicated to N. N. Uraltseva on her jubilee
Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 221, 1995, pp. 145–166.
Translated by S. Yu. Pilyugin.
Rights and permissions
About this article
Cite this article
Malyshev, V.A. The Bergman kernel and the green function. J Math Sci 87, 3366–3380 (1997). https://doi.org/10.1007/BF02355588
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF02355588