On Bell's duality theorem for harmonic functions

Motos, Joaquín; Pérez-Esteva, Salvador

Studia Mathematica, Tome 133 (1999), p. 49-60 / Harvested from The Polish Digital Mathematics Library

Résumé

Define $h^{\infty} (E)$ as the subspace of $C^{\infty} (B ̅ L, E)$ 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 $h^{- \infty} (E *)$ consisting of all harmonic E*-valued functions g such that ${(1 - | x |)}^{m} f$ is bounded for some m>0. Then the dual $h^{\infty} (E *)$ is represented by $h^{- \infty} (E *)$ through $⟨ f, g ⟩_{0} = l i m_{r \to 1} ʃ_{B} ⟨ f (r x), g (x) ⟩ d x$ , $f \in h^{- \infty} (E *), g \in h^{\infty} (E)$ . This extends the results of S. Bell in the scalar case.

Publié le : 1999-01-01

Zbl 0957.46017

EUDML-ID : urn:eudml:doc:216674

@article{bwmeta1.element.bwnjournal-article-smv137i1p49bwm,
     author = {Joaqu\'\i n Motos and Salvador P\'erez-Esteva},
     title = {On Bell's duality theorem for harmonic functions},
     journal = {Studia Mathematica},
     volume = {133},
     year = {1999},
     pages = {49-60},
     zbl = {0957.46017},
     language = {en},
     url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-smv137i1p49bwm}
}

Motos, Joaquín; Pérez-Esteva, Salvador. On Bell's duality theorem for harmonic functions. Studia Mathematica, Tome 133 (1999) pp. 49-60. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-smv137i1p49bwm/

Bibliographie

[00000] [1] R. A. Adams, Sobolev Spaces, Academic Press, 1975.

[00001] [2] S. Bell, A duality theorem for harmonic functions, Michigan Math. J. 29 (1982), 123-128. | Zbl 0482.31004

[00002] [3] O. Blasco, Boundary values of vector-valued harmonic functions considered as operators, Studia Math. 86 (1987), 19-33. | Zbl 0639.46047

[00003] [4] O. Blasco, Boundary values of functions in vector-valued Hardy spaces and geometry on Banach spaces, J. Funct. Anal. 78 (1988), 346-364. | Zbl 0658.46031

[00004] [5] O. Blasco and S. Pérez-Esteva, $L^{p}$ continuity of projectors of weighted harmonic Bergman spaces, preprint. | Zbl 0947.31001

[00005] [6] R. Coifman and R. Rochberg, Representation theorems for holomorphic and harmonic functions in $L^{p}$ , Astérisque 77 (1980), 11-66. | Zbl 0472.46040

[00006] [7] J. Diestel and J. J. Uhl, Vector Measures, Amer. Math. Soc., 1977.

[00007] [8] J. Dieudonné, Foundations of Modern Analysis, Academic Press, New York, 1960. | Zbl 0100.04201

[00008] [9] N. Dinculeanu, Vector Measures, Pergamon Press, New York, 1967.

[00009] [10] A. E. Djrbashian and F. A. Shamoian, Topics in the Theory of $A_{α}^{p}$ Spaces, Teubner-Texte zur Math., 1988.

[00010] [11] A. Grothendieck, Produits tensoriels et espaces nucléaires, Mem. Amer. Math. Soc. 16 (1955).

[00011] [12] G. Köthe, Topological Vector Spaces I, II, Springer, Heidelberg, 1969, 1979. | Zbl 0179.17001

[00012] [13] E. Ligocka, The Sobolev spaces of harmonic functions, Studia Math. 84 (1986), 79-87. | Zbl 0627.46033

[00013] [14] E. Ligocka, Estimates in Sobolev norms $∥ \cdot ∥_{p}^{s}$ for harmonic and holomorphic functions and interpolation between Sobolev and Hölder spaces of harmonic functions, ibid. 86 (1987), 255-271. | Zbl 0642.46035

[00014] [15] E. Ligocka, On the space of Bloch harmonic functions and interpolation of spaces of harmonic and holomorphic functions, ibid. 87 (1987), 223-238. | Zbl 0657.31006

[00015] [16] S. Pérez-Esteva, Duality on vector-valued weighted harmonic Bergman spaces, ibid. 118 (1996), 37-47. | Zbl 0854.46022

[00016] [17] E. Straube, Harmonic and analytic functions admitting a distribution boundary value, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 11 (1984), 559-591. | Zbl 0582.31003