# "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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topBonilla, 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 -

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