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

Colloquium Mathematicae (2000)

• Volume: 83, Issue: 2, page 155-160
• ISSN: 0010-1354

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## Abstract

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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‖μDαv(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 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 .

## How to cite

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Bonilla, A.. ""Counterexamples" to the harmonic Liouville theorem and harmonic functions with zero nontangential limits." Colloquium Mathematicae 83.2 (2000): 155-160. <http://eudml.org/doc/210777>.

@article{Bonilla2000,
abstract = {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‖μDαv(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 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 .},
author = {Bonilla, A.},
journal = {Colloquium Mathematicae},
keywords = {nontangential limits; universal function; approximation; Liouville harmonic theorem; Radon transform; harmonic functions; nontangential limit; harmonic approximation on non-compact sets},
language = {eng},
number = {2},
pages = {155-160},
title = {"Counterexamples" to the harmonic Liouville theorem and harmonic functions with zero nontangential limits},
url = {http://eudml.org/doc/210777},
volume = {83},
year = {2000},
}

TY - JOUR
AU - Bonilla, A.
TI - "Counterexamples" to the harmonic Liouville theorem and harmonic functions with zero nontangential limits
JO - Colloquium Mathematicae
PY - 2000
VL - 83
IS - 2
SP - 155
EP - 160
AB - 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‖μDαv(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 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 .
LA - eng
KW - nontangential limits; universal function; approximation; Liouville harmonic theorem; Radon transform; harmonic functions; nontangential limit; harmonic approximation on non-compact sets
UR - http://eudml.org/doc/210777
ER -

## References

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8. [8] S. Axler, P. Bourdon and W. Ramsey, Harmonic Function Theory, Springer, New York, 1992. Zbl0765.31001
9. [9] L. Bernal González, A lot of 'counterexamples' to Liouville's theorem, J. Math. Anal. Appl. 201 (1996), 1002-1009. Zbl0855.30030
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11. [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. Zbl0831.30024

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