The Bergman kernel functions of certain unbounded domains

• Autorzy: Friedrich Haslinger
• Języki publikacji: EN

Abstrakty

EN

We compute the Bergman kernel functions of the unbounded domains $Ω_p = {(z',z) ∈ ℂ² : 𝕴z > p(z')}$, where $p(z') = |z'|^{α}/α$. It is also shown that these kernel functions have no zeros in $Ω_p$. We use a method from harmonic analysis to reduce the computation of the 2-dimensional case to the problem of finding the kernel function of a weighted space of entire functions in one complex variable.

Słowa kluczowe

EN

Bergman kernel   Szegő kernel  

Kategorie tematyczne

• 32A07: Special domains (Reinhardt, Hartogs, circular, tube)
• 32A15: Entire functions

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Annales Polonici Mathematici
• Rocznik: 1998
• Tom: 70
• Numer: 1
• Strony: 109-115

Daty

wydano

1998

otrzymano

1997-12-29

poprawiono

1998-08-14

Twórcy

autor

Friedrich Haslinger

• Institut für Mathematik, Universität Wien, Strudlhofgasse 4, A-1090 Wien, Austria

Bibliografia

• [BFS] H. P. Boas, S. Fu and E. J. Straube, The Bergman kernel function: Explicit formulas and zeros, Proc. Amer. Math. Soc. (to appear).
• [BL] A. Bonami and N. Lohoué, Projecteurs de Bergman et Szegő pour une classe de domaines faiblement pseudo-convexes et estimations $L^p$, Compositio Math. 46 (1982), 159-226.
• [BSY] H. P. Boas, E. J. Straube and J. Yu, Boundary limits of the Bergman kernel and metric, Michigan Math. J. 42 (1995), 449-462.
• [D'A] J. P. D'Angelo, An explicit computation of the Bergman kernel function, J. Geom. Anal. 4 (1994), 23-34.
• [D] K. P. Diaz, The Szegő kernel as a singular integral kernel on a family of weakly pseudoconvex domains, Trans. Amer. Math. Soc. 304 (1987), 147-170.
• [DO] K. Diederich and T. Ohsawa, On the parameter dependence of solutions to the ∂̅-equation, Math. Ann. 289 (1991), 581-588.
• [FH1] G. Francsics and N. Hanges, Explicit formulas for the Szegő kernel on certain weakly pseudoconvex domains, Proc. Amer. Math. Soc. 123 (1995), 3161-3168.
• [FH2] G. Francsics and N. Hanges, The Bergman kernel of complex ovals and multivariable hypergeometric functions, J. Funct. Anal. 142 (1996), 494-510.
• [FH3] G. Francsics and N. Hanges, Asymptotic behavior of the Bergman kernel and hypergeometric functions, in: Contemp. Math. (to appear).
• [GS] P. C. Greiner and E. M. Stein, On the solvability of some differential operators of type $⎕_b$, Proc. Internat. Conf., (Cortona, 1976-1977), Scuola Norm. Sup. Pisa, 1978, 106-165.
• [Han] N. Hanges, Explicit formulas for the Szegő kernel for some domains in $ℂ^2$, J. Funct. Anal. 88 (1990), 153-165.
• [Has1] F. Haslinger, Szegő kernels of certain unbounded domains in $ℂ^2$, Rév. Rou- maine Math. Pures Appl. 39 (1994), 914-926.
• [Has2] F. Haslinger, Singularities of the Szegő kernel for certain weakly pseudoconvex domains in $ℂ^2$, J. Funct. Anal. 129 (1995), 406-427.
• [Has3] F. Haslinger, Bergman and Hardy spaces on model domains, Illinois J. Math. (to appear).
• [He] P. Henrici, Applied and Computational Complex Analysis, II, Wiley, New York, 1977.
• [K] H. Kang, $∂̅_b$-equations on certain unbounded weakly pseudoconvex domains, Trans. Amer. Math. Soc. 315 (1989), 389-413.
• [Kr] S. G. Krantz, Function Theory of Several Complex Variables, Wadsworth & Brooks/Cole, Pacific Grove, Calif., 1992.
• [McN1] J. McNeal, Boundary behavior of the Bergman kernel function in $ℂ^2$, Duke Math. J. 58 (1989), 499-512.
• [McN2] J. McNeal, Local geometry of decoupled pseudoconvex domains, in: Complex Analysis, Aspects of Math. E17, K. Diederich (ed.), Vieweg 1991, 223-230.
• [McN3] J. McNeal, Estimates on the Bergman kernels on convex domains, Adv. Math. 109 (1994), 108-139.
• [N] A. Nagel, Vector fields and nonisotropic metrics, in: Beijing Lectures in Harmonic Analysis, E. M. Stein (ed.), Princeton Univ. Press, Princeton, N.J., 1986, 241-306.
• [NRSW1] A. Nagel, J. P. Rosay, E. M. Stein and S. Wainger, Estimates for the Bergman and Szegő kernels in certain weakly pseudoconvex domains, Bull. Amer. Math. Soc. 18 (1988), 55-59.
• [NRSW2] A. Nagel, J. P. Rosay, E. M. Stein and S. Wainger, Estimates for the Bergman and Szegő kernels in $ℂ^2$, Ann. of Math. 129 (1989), 113-149.
• [OPY] K. Oeljeklaus, P. Pflug and E. H. Youssfi, The Bergman kernel of the minimal ball and applications, Ann. Inst. Fourier (Grenoble) 47 (1997), 915-928.
• [R] M. Range, Holomorphic Functions and Integral Representations in Several Complex Variables, Springer, 1986.
• [S] E. M. Stein, Harmonic Analysis. Real-Variable Methods, Orthogonality and Oscillatory Integrals, Princeton Univ. Press, Princeton, N.J., 1993.
• [T] B. A. Taylor, On weighted polynomial approximation of entire functions, Pacific J. Math. 36 (1971), 523-539.