# Duality on vector-valued weighted harmonic Bergman spaces

Studia Mathematica (1996)

- Volume: 118, Issue: 1, page 37-47
- ISSN: 0039-3223

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topPérez-Esteva, Salvador. "Duality on vector-valued weighted harmonic Bergman spaces." Studia Mathematica 118.1 (1996): 37-47. <http://eudml.org/doc/216262>.

@article{Pérez1996,

abstract = {We study the duals of the spaces $A^\{pα\}(X)$ of harmonic functions in the unit ball of $ℝ^n$ with values in a Banach space X, belonging to the Bochner $L^p$ space with weight $(1-|x|)^α$, denoted by $L^\{pα\}(X)$. For 0 < α < p-1 we construct continuous projections onto $A^\{pα\}(X)$ providing a decomposition $L^\{pα\}(X) = A^\{pα\}(X) + M^\{pα\}(X)$. We discuss the conditions on p, α and X for which $A^\{pα\}(X)* = A^\{qα\}(X*)$ and $M^\{pα\}(X)* = M^\{qα\}(X*)$, 1/p+1/q = 1. The last equality is equivalent to the Radon-Nikodým property of X*.},

author = {Pérez-Esteva, Salvador},

journal = {Studia Mathematica},

keywords = {vector-valued weighted harmonic Bergman spaces; duals; harmonic functions; Bochner space; continuous projections; Radon-Nikodým property},

language = {eng},

number = {1},

pages = {37-47},

title = {Duality on vector-valued weighted harmonic Bergman spaces},

url = {http://eudml.org/doc/216262},

volume = {118},

year = {1996},

}

TY - JOUR

AU - Pérez-Esteva, Salvador

TI - Duality on vector-valued weighted harmonic Bergman spaces

JO - Studia Mathematica

PY - 1996

VL - 118

IS - 1

SP - 37

EP - 47

AB - We study the duals of the spaces $A^{pα}(X)$ of harmonic functions in the unit ball of $ℝ^n$ with values in a Banach space X, belonging to the Bochner $L^p$ space with weight $(1-|x|)^α$, denoted by $L^{pα}(X)$. For 0 < α < p-1 we construct continuous projections onto $A^{pα}(X)$ providing a decomposition $L^{pα}(X) = A^{pα}(X) + M^{pα}(X)$. We discuss the conditions on p, α and X for which $A^{pα}(X)* = A^{qα}(X*)$ and $M^{pα}(X)* = M^{qα}(X*)$, 1/p+1/q = 1. The last equality is equivalent to the Radon-Nikodým property of X*.

LA - eng

KW - vector-valued weighted harmonic Bergman spaces; duals; harmonic functions; Bochner space; continuous projections; Radon-Nikodým property

UR - http://eudml.org/doc/216262

ER -

## References

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- [7] E. Ligocka, The Hölder duality for harmonic functions, Studia Math. 84 (1986), 269-277. Zbl0651.46035
- [8] E. Ligocka, Estimates in Sobolev norms $\parallel \xb7{\parallel}_{p}^{s}$ for harmonic and holomorphic functions and interpolation between Sobolev and Hölder spaces of harmonic functions, Studia Math. 86 (1987), 255-271. Zbl0642.46035
- [9] E. Ligocka, On the reproducing kernel for harmonic functions and the space of Bloch harmonic functions on the unit ball in ${R}^{n}$, ibid. 87 (1987), 23-32. Zbl0658.31006
- [10] C. B. Morrey, Multiple Integrals in the Calculus of Variations, Springer, New York, 1966. Zbl0142.38701
- [11] E. M. Stein and G. Weiss, Introduction to Fourier Analysis on Euclidean Spaces, Princeton University Press, 1971. Zbl0232.42007

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