Duality on vector-valued weighted harmonic Bergman spaces

Salvador Pérez-Esteva

Studia Mathematica (1996)

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

Abstract

top
We study the duals of the spaces A p α ( 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 p α ( X ) . For 0 < α < p-1 we construct continuous projections onto A p α ( X ) providing a decomposition L p α ( X ) = A p α ( X ) + M p α ( X ) . We discuss the conditions on p, α and X for which A p α ( X ) * = A q α ( X * ) and M p α ( X ) * = M q α ( X * ) , 1/p+1/q = 1. The last equality is equivalent to the Radon-Nikodým property of X*.

How to cite

top

Pérez-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

top
  1. [1] S. Bell, A duality theorem for harmonic functions, Michigan Math. J. 29 (1982), 123-128. Zbl0482.31004
  2. [2] C. V. Coffman and J. Cohen, The duals of harmonic Bergman spaces, Proc. Amer. Math. Soc. 110 (1990), 697-704. Zbl0717.31001
  3. [3] R. R. Coifman and R. Rochberg, Representation theorems for holomorphic and harmonic functions in L p , Astérisque 77 (1980), 11-66. Zbl0472.46040
  4. [4] J. Diestel and J. J. Uhl, Vector Measures, Math. Surveys 15, Amer. Math. Soc., Providence, R.I., 1977. 
  5. [5] N. Dinculeanu, Vector Measures, Pergamon Press, New York, 1967. 
  6. [6] J. García-Cuerva and J. L. Rubio de Francia, Weighted Norm Inequalities and Related Topics, Notas Mat. 116, North-Holland, Amsterdam, 1985. 
  7. [7] E. Ligocka, The Hölder duality for harmonic functions, Studia Math. 84 (1986), 269-277. Zbl0651.46035
  8. [8] E. Ligocka, Estimates in Sobolev norms ∥ · ∥ 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. [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. [10] C. B. Morrey, Multiple Integrals in the Calculus of Variations, Springer, New York, 1966. Zbl0142.38701
  11. [11] E. M. Stein and G. Weiss, Introduction to Fourier Analysis on Euclidean Spaces, Princeton University Press, 1971. Zbl0232.42007

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.