Warianty tytułu
Języki publikacji
Abstrakty
In this paper we consider spaces of weight square-integrable and harmonic functions L^2H(Ω,μ). Weights μ for which there exists reproducing kernel of L^2H(Ω,μ) are named 'admissible weights' and such kernels are named 'harmonic Bergman kernels'. We prove that if only weight of integration is integrable in some negative power, then it is admissible. Next we construct a weight μ on the unit circle which is non-admissible and using Bell-Ligocka theorem we show that such weights exist for a large class of domains in ℝ^2. Later we conclude from the classical result of reproducing kernel Hilbert spaces theory that if the set {f∈L^2H(Ω,μ)| f(z) = c} for admissible weight μ is non-empty, then there is exactly one element with minimal norm. Such an element in this paper is called 'a minimal (z,c)-solution in weight μ of Laplace's equation on Ω' and upper estimates for it are given.(original abstract)
Słowa kluczowe
Twórcy
autor
- Military University of Technology, Warsaw, Poland
Bibliografia
- Axler S., Bourdon P., and Ramey W., Harmonic Function Theory, Springer-Verlag, New York, 2001.
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- Figueroa R. and López Pouso R., Minimal and maximal solutions to first-order differential equations with state-dependent deviated arguments, Bound. Value Probl. 2012, 2012:7, 12 pp.
- Gipple J., The volume of n-balls, Rose-Hulman Undergrad. Math. J. 15 (2014), no. 1, 237-248.
- Hörmander L., An Introduction to Complex Analysis in Several Variables, North-Holland Publishing Co., Amsterdam, 1990.
- Kang H. and Koo H., Estimates of the harmonic Bergman kernel on smooth domains, J. Funct. Anal. 185 (2001), no. 1, 220-239.
- Koo H. and Nam K., Yi H., Weighted harmonic Bergman kernel on half-spaces, J. Math. Soc. Japan 58 (2006), no. 2, 351-362.
- Krantz S.G., Function Theory of Several Complex Variables, AMS Chelsea Publishing, Providence, RI, 2001.
- Pasternak-Winiarski Z., On weights which admit the reproducing kernel of Bergman type, Internat. J. Math. Math. Sci. 15 (1992), no. 1, 1-14.
- Ramey W.C. and Yi H., Harmonic Bergman functions on half-spaces, Trans. Amer. Math. Soc. 348 (1996), no. 2, 633-660.
- Rudin W., Real and Complex Analysis, McGraw-Hill Book Co., New York, 1974.
- Troianiello G.M., Maximal and minimal solutions to a class of elliptic quasilinear problems, Proc. Amer. Math. Soc. 91 (1984), no. 1, 95-101.
Typ dokumentu
Bibliografia
Identyfikatory
Identyfikator YADDA
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