On weights which admit harmonic Bergman kernel and minimal solutions of Laplace's equation
Abstract
In this paper we consider spaces of weight square-integrable and harmonic functions L2H(Ω,μ). Weights μ for which there exists reproducing kernel of L2H(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 ℝ2. Later we conclude from the classical result of reproducing kernel Hilbert spaces theory that if the set {f∈L2H(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.
Keywords
harmonic Bergman kernel; reproducing kernel Hilbert space; weight of integration; admissible weight; functional of point evaluation; Laplace’s equation; minimal solution
References
S. Axler, P. Bourdon, and W. Ramey, Harmonic Function Theory, Springer-Verlag, New York, 2001.
S. Bell and E. Ligocka, A simplification and extension of Fefferman’s theorem on biholomorphic mappings, Invent. Math. 57 (1980), no. 3, 283–289.
R. Figueroa and R. López Pouso, Minimal and maximal solutions to first-order differential equations with state-dependent deviated arguments, Bound. Value Probl. 2012, 2012:7, 12 pp.
J. Gipple, The volume of n-balls, Rose-Hulman Undergrad. Math. J. 15 (2014), no. 1, 237–248.
L. Hörmander, An Introduction to Complex Analysis in Several Variables, North-Holland Publishing Co., Amsterdam, 1990.
H. Kang and H. Koo, Estimates of the harmonic Bergman kernel on smooth domains, J. Funct. Anal. 185 (2001), no. 1, 220–239.
H. Koo and K. Nam, H. Yi, Weighted harmonic Bergman kernel on half-spaces, J. Math. Soc. Japan 58 (2006), no. 2, 351–362.
S.G. Krantz, Function Theory of Several Complex Variables, AMS Chelsea Publishing, Providence, RI, 2001.
Z. Pasternak-Winiarski, On weights which admit the reproducing kernel of Bergman type, Internat. J. Math. Math. Sci. 15 (1992), no. 1, 1–14.
W.C. Ramey and H. Yi, Harmonic Bergman functions on half-spaces, Trans. Amer. Math. Soc. 348 (1996), no. 2, 633–660.
W. Rudin, Real and Complex Analysis, McGraw-Hill Book Co., New York, 1974.
G.M. Troianiello, Maximal and minimal solutions to a class of elliptic quasilinear problems, Proc. Amer. Math. Soc. 91 (1984), no. 1, 95–101.
Wydział Cybernetyki, Wojskowa Akademia Techniczna w Warszawie Poland
https://orcid.org/0000-0003-1813-0519
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