Harmonic functions on the real hyperbolic ball I: Boundary values and atomic decomposition of Hardy spaces

Philippe Jaming

Colloquium Mathematicae (1999)

  • Volume: 80, Issue: 1, page 63-82
  • ISSN: 0010-1354

Abstract

top
We study harmonic functions for the Laplace-eltrami operator on the real hyperbolic space n . We obtain necessary and sufficient conditions for these functions and their normal derivatives to have a boundary distribution. In doing so, we consider different behaviors of hyperbolic harmonic functions according to the parity of the dimension of the hyperbolic ball n . We then study the Hardy spaces H p ( n ) , 0

How to cite

top

Jaming, Philippe. "Harmonic functions on the real hyperbolic ball I: Boundary values and atomic decomposition of Hardy spaces." Colloquium Mathematicae 80.1 (1999): 63-82. <http://eudml.org/doc/210706>.

@article{Jaming1999,
abstract = {We study harmonic functions for the Laplace-eltrami operator on the real hyperbolic space $_n$. We obtain necessary and sufficient conditions for these functions and their normal derivatives to have a boundary distribution. In doing so, we consider different behaviors of hyperbolic harmonic functions according to the parity of the dimension of the hyperbolic ball $_n$. We then study the Hardy spaces $H^p(_n)$, 0},
author = {Jaming, Philippe},
journal = {Colloquium Mathematicae},
keywords = {Hardy spaces; atomic decomposition; real hyperbolic ball; boundary values; harmonic functions; Laplace-Beltrami operator},
language = {eng},
number = {1},
pages = {63-82},
title = {Harmonic functions on the real hyperbolic ball I: Boundary values and atomic decomposition of Hardy spaces},
url = {http://eudml.org/doc/210706},
volume = {80},
year = {1999},
}

TY - JOUR
AU - Jaming, Philippe
TI - Harmonic functions on the real hyperbolic ball I: Boundary values and atomic decomposition of Hardy spaces
JO - Colloquium Mathematicae
PY - 1999
VL - 80
IS - 1
SP - 63
EP - 82
AB - We study harmonic functions for the Laplace-eltrami operator on the real hyperbolic space $_n$. We obtain necessary and sufficient conditions for these functions and their normal derivatives to have a boundary distribution. In doing so, we consider different behaviors of hyperbolic harmonic functions according to the parity of the dimension of the hyperbolic ball $_n$. We then study the Hardy spaces $H^p(_n)$, 0
LA - eng
KW - Hardy spaces; atomic decomposition; real hyperbolic ball; boundary values; harmonic functions; Laplace-Beltrami operator
UR - http://eudml.org/doc/210706
ER -

References

top
  1. [1] P. Ahern, J. Bruna and C. Cascante, H p -theory for generalized m a t h c a l M -harmonic functions in the unit ball, Indiana Univ. Math. J. 45 (1996), 103-145. Zbl0866.32007
  2. [2] E. P. van den Ban and H. Schlichtkrull, Assymptotic expansions and boundary values of eigenfunctions on Riemannian symmetric spaces, J. Reine Angew. Math. 380 (1987), 108-165. Zbl0631.58028
  3. [3] A. Bonami, J. Bruna and S. Grellier, On Hardy, BMO and Lipschitz spaces of invariant harmonic functions in the unit ball, Proc. London Math. Soc. 77 (1998), 665-696. Zbl0904.31007
  4. [4] L. Colzani, Hardy spaces on unit spheres, Boll. Un. Mat. Ital. C (6) 4 (1985), 219-244. Zbl0585.42021
  5. [5] A. Erdélyi et al. (eds.), Higher Transcendental Functions I, McGraw-Hill, 1953. Zbl0051.30303
  6. [6] C. Fefferman and E. M. Stein, H p spaces of several variables, Acta Math. 129 (1972), 137-193. Zbl0257.46078
  7. [7] J. B. Garnett and R. H. Latter, The atomic decomposition for Hardy spaces in several complex variables, Duke Math. J. 45 (1978), 815-845. Zbl0403.32006
  8. [8] P. Jaming, Trois problèmes d'analyse harmonique, PhD thesis, Université d'Orlé- ans, 1998. 
  9. [9] S. G. Krantz and S. Y. Li, On decomposition theorems for Hardy spaces on domains in n and applications, J. Fourier Anal. Appl. 2 (1995), 65-107. Zbl0886.32003
  10. [10] J. B. Lewis, Eigenfunctions on symmetric spaces with distribution-valued boundary forms, J. Funct. Anal. 29 (1978), 287-307. Zbl0398.43010
  11. [11] K. Minemura, Harmonic functions on real hyperbolic spaces, Hiroshima Math. J. 3 (1973), 121-151. Zbl0274.31004
  12. [12] K. Minemura, Eigenfunctions of the Laplacian on a real hyperbolic spaces, J. Math. Soc. Japan 27 (1975), 82-105. Zbl0292.35023
  13. [13] H. Samii, Les transformations de Poisson dans la boule hyperbolique, PhD thesis, Université Nancy 1, 1982. 
  14. [14] E. M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton Univ. Press, 1970. Zbl0207.13501

NotesEmbed ?

top

You must be logged in to post comments.