We prove that, if μ>0, then there exists a linear manifold M of harmonic functions in which is dense in the space of all harmonic functions in 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 , 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 .
@article{bwmeta1.element.bwnjournal-article-cmv83i2p155bwm,
author = {A. Bonilla},
title = {"Counterexamples" to the harmonic Liouville theorem and harmonic functions with zero nontangential limits},
journal = {Colloquium Mathematicae},
volume = {84/85},
year = {2000},
pages = {155-160},
zbl = {0967.31001},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-cmv83i2p155bwm}
}
Bonilla, A. "Counterexamples" to the harmonic Liouville theorem and harmonic functions with zero nontangential limits. Colloquium Mathematicae, Tome 84/85 (2000) pp. 155-160. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-cmv83i2p155bwm/
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