# On Bell's duality theorem for harmonic functions

Studia Mathematica (1999)

• Volume: 137, Issue: 1, page 49-60
• ISSN: 0039-3223

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## Abstract

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Define ${h}^{\infty }\left(E\right)$ as the subspace of ${C}^{\infty }\left(B̅L,E\right)$ consisting of all harmonic functions in B, where B is the ball in the n-dimensional Euclidean space and E is any Banach space. Consider also the space ${h}^{-\infty }\left(E*\right)$ consisting of all harmonic E*-valued functions g such that ${\left(1-|x|\right)}^{m}f$ is bounded for some m>0. Then the dual ${h}^{\infty }\left(E*\right)$ is represented by ${h}^{-\infty }\left(E*\right)$ through $⟨f,g{⟩}_{0}=li{m}_{r\to 1}{ʃ}_{B}⟨f\left(rx\right),g\left(x\right)⟩dx$, $f\in {h}^{-\infty }\left(E*\right),g\in {h}^{\infty }\left(E\right)$. This extends the results of S. Bell in the scalar case.

## How to cite

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Motos, Joaquín, and Pérez-Esteva, Salvador. "On Bell's duality theorem for harmonic functions." Studia Mathematica 137.1 (1999): 49-60. <http://eudml.org/doc/216674>.

@article{Motos1999,
abstract = {Define $h^∞(E)$ as the subspace of $C^∞(B̅L,E)$ consisting of all harmonic functions in B, where B is the ball in the n-dimensional Euclidean space and E is any Banach space. Consider also the space $h^\{-∞\}(E*)$ consisting of all harmonic E*-valued functions g such that $(1-|x|)^mf$ is bounded for some m>0. Then the dual $h^∞(E*)$ is represented by $h^\{-∞\}(E*)$ through $⟨f,g⟩_0= lim_\{r→1\}ʃ_B ⟨f(rx),g(x)⟩dx$, $f ∈ h^\{-∞\}(E*),g ∈ h^∞(E)$. This extends the results of S. Bell in the scalar case.},
author = {Motos, Joaquín, Pérez-Esteva, Salvador},
journal = {Studia Mathematica},
keywords = {harmonic functions; dual space; vector-valued Sobolev spaces; Bell's duality theorem},
language = {eng},
number = {1},
pages = {49-60},
title = {On Bell's duality theorem for harmonic functions},
url = {http://eudml.org/doc/216674},
volume = {137},
year = {1999},
}

TY - JOUR
AU - Motos, Joaquín
AU - Pérez-Esteva, Salvador
TI - On Bell's duality theorem for harmonic functions
JO - Studia Mathematica
PY - 1999
VL - 137
IS - 1
SP - 49
EP - 60
AB - Define $h^∞(E)$ as the subspace of $C^∞(B̅L,E)$ consisting of all harmonic functions in B, where B is the ball in the n-dimensional Euclidean space and E is any Banach space. Consider also the space $h^{-∞}(E*)$ consisting of all harmonic E*-valued functions g such that $(1-|x|)^mf$ is bounded for some m>0. Then the dual $h^∞(E*)$ is represented by $h^{-∞}(E*)$ through $⟨f,g⟩_0= lim_{r→1}ʃ_B ⟨f(rx),g(x)⟩dx$, $f ∈ h^{-∞}(E*),g ∈ h^∞(E)$. This extends the results of S. Bell in the scalar case.
LA - eng
KW - harmonic functions; dual space; vector-valued Sobolev spaces; Bell's duality theorem
UR - http://eudml.org/doc/216674
ER -

## References

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13. [13] E. Ligocka, The Sobolev spaces of harmonic functions, Studia Math. 84 (1986), 79-87. Zbl0627.46033
14. [14] E. Ligocka, Estimates in Sobolev norms $\parallel ·{\parallel }_{p}^{s}$ for harmonic and holomorphic functions and interpolation between Sobolev and Hölder spaces of harmonic functions, ibid. 86 (1987), 255-271. Zbl0642.46035
15. [15] E. Ligocka, On the space of Bloch harmonic functions and interpolation of spaces of harmonic and holomorphic functions, ibid. 87 (1987), 223-238. Zbl0657.31006
16. [16] S. Pérez-Esteva, Duality on vector-valued weighted harmonic Bergman spaces, ibid. 118 (1996), 37-47. Zbl0854.46022
17. [17] E. Straube, Harmonic and analytic functions admitting a distribution boundary value, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 11 (1984), 559-591. Zbl0582.31003

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