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2000 | 83 | 2 | 155-160
Tytuł artykułu

"Counterexamples" to the harmonic Liouville theorem and harmonic functions with zero nontangential limits

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Treść / Zawartość
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Języki publikacji
EN
Abstrakty
EN
We prove that, if μ>0, then there exists a linear manifold M of harmonic functions in $ℝ^N$ which is dense in the space of all harmonic functions in $ℝ^N$ and lim_{{‖x‖→∞} {x ∈ S}} ‖x‖^{μ}D^{α}v(x) = 0 for every v ∈ M and multi-index α, where S denotes any hyperplane strip. Moreover, every nonnull function in M is universal. In particular, if μ ≥ N+1, then every function v ∈ M satisfies ∫_H vdλ =0 for every (N-1)-dimensional hyperplane H, where λ denotes the (N-1)-dimensional Lebesgue measure. On the other hand, we prove that there exists a linear manifold M of harmonic functions in the unit ball 𝔹 of $ℝ^N$, which is dense in the space of all harmonic functions and each function in M has zero nontangential limit at every point of the boundary of 𝔹.
Rocznik
Tom
83
Numer
2
Strony
155-160
Opis fizyczny
Daty
wydano
2000
otrzymano
1999-03-15
Twórcy
autor
  • Departamento de Análisis Matemático, Universidad de La Laguna, 38271 La Laguna (Tenerife), Canary Islands, Spain
Bibliografia
  • [1] D. H. Armitage, A non-constant continuous function on the plane whose integral on every line is zero, Amer. Math. Monthly 101 (1994), 892-894.
  • [2] D. H. Armitage and P. M. Gauthier, Recent developments in harmonic approximation, with applications, Results Math. 29 (1996), 1-15.
  • [3] D. H. Armitage and M. Goldstein, Better than uniform approximation on closed sets by harmonic functions with singularities, Proc. London Math. Soc. 60 (1990), 319-343.
  • [4] D. H. Armitage and M. Goldstein, Radial limiting behavior of harmonic functions in cones, Complex Variables 22 (1993), 267-276.
  • [5] D. H. Armitage and M. Goldstein, Nonuniqueness for the Radon transform, Proc. Amer. Math. Soc. 117 (1993), 175-178.
  • [6] D. H. Armitage and M. Goldstein, Tangential harmonic approximation on relatively closed sets, Proc. London Math. Soc. 68 (1994), 112-126.
  • [7] J. M. Ash and R. Brown, Uniqueness and nonuniqueness for harmonic functions with zero nontangential limits, Harmonic Analysis (Sendai, 1990), ICM-90 Satell. Conf. Proc., S. Igari (ed.), Springer, 1991, 30-40.
  • [8] S. Axler, P. Bourdon and W. Ramsey, Harmonic Function Theory, Springer, New York, 1992.
  • [9] L. Bernal González, A lot of 'counterexamples' to Liouville's theorem, J. Math. Anal. Appl. 201 (1996), 1002-1009.
  • [10] L. Bernal González, Small entire functions with extremely fast growth, ibid. 207 (1997), 541-548.
  • [11] L. Bernal González and A. Montes Rodríguez, Non-finite dimensional closed vector space of universal functions for composition operators, J. Approx. Theory 82 (1995), 375-391.
Typ dokumentu
Bibliografia
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bwmeta1.element.bwnjournal-article-cmv83i2p155bwm
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