Dirichlet forms on symmetric spaces

Christian Berg

Annales de l'institut Fourier (1973)

  • Volume: 23, Issue: 1, page 135-156
  • ISSN: 0373-0956

Abstract

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Let G be a locally compact group and K a compact subgroup such that the algebra L 1 ( G ) ♯ of biinvariant integrable functions is commutative. We characterize the G -invariant Dirichlet forms on the homogeneous space G / K using harmonic analysis of L 1 ( G ) ♯ . This extends results from Ch. Berg, Séminaire Brelot-Choquet-Deny, Paris, 13e année 1969/70 and J. Deny, Potential theory (C.I.M.E., I ciclo, Stresa), Ed. Cremonese, Rome, 1970. Every non-zero G -invariant Dirichlet form on a symmetric space G / K of non compact type of rank one give rise to a regular Dirichlet space, and these potentials of finite energy are square integrable in contrast to euclidean space.

How to cite

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Berg, Christian. "Dirichlet forms on symmetric spaces." Annales de l'institut Fourier 23.1 (1973): 135-156. <http://eudml.org/doc/74107>.

@article{Berg1973,
abstract = {Let $G$ be a locally compact group and $K$ a compact subgroup such that the algebra $L^1(G)^\sharp $ of biinvariant integrable functions is commutative. We characterize the $G$-invariant Dirichlet forms on the homogeneous space $G/K$ using harmonic analysis of $L^1(G)^\sharp $. This extends results from Ch. Berg, Séminaire Brelot-Choquet-Deny, Paris, 13e année 1969/70 and J. Deny, Potential theory (C.I.M.E., I ciclo, Stresa), Ed. Cremonese, Rome, 1970. Every non-zero $G$-invariant Dirichlet form on a symmetric space $G/K$ of non compact type of rank one give rise to a regular Dirichlet space, and these potentials of finite energy are square integrable in contrast to euclidean space.},
author = {Berg, Christian},
journal = {Annales de l'institut Fourier},
language = {eng},
number = {1},
pages = {135-156},
publisher = {Association des Annales de l'Institut Fourier},
title = {Dirichlet forms on symmetric spaces},
url = {http://eudml.org/doc/74107},
volume = {23},
year = {1973},
}

TY - JOUR
AU - Berg, Christian
TI - Dirichlet forms on symmetric spaces
JO - Annales de l'institut Fourier
PY - 1973
PB - Association des Annales de l'Institut Fourier
VL - 23
IS - 1
SP - 135
EP - 156
AB - Let $G$ be a locally compact group and $K$ a compact subgroup such that the algebra $L^1(G)^\sharp $ of biinvariant integrable functions is commutative. We characterize the $G$-invariant Dirichlet forms on the homogeneous space $G/K$ using harmonic analysis of $L^1(G)^\sharp $. This extends results from Ch. Berg, Séminaire Brelot-Choquet-Deny, Paris, 13e année 1969/70 and J. Deny, Potential theory (C.I.M.E., I ciclo, Stresa), Ed. Cremonese, Rome, 1970. Every non-zero $G$-invariant Dirichlet form on a symmetric space $G/K$ of non compact type of rank one give rise to a regular Dirichlet space, and these potentials of finite energy are square integrable in contrast to euclidean space.
LA - eng
UR - http://eudml.org/doc/74107
ER -

References

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  1. [1] Chr. BERG, Suites définies négatives et espaces de Dirichlet sur la sphère, Séminaire Brelot-Choquet-Deny, Paris, 13e année 1969/1970. Zbl0213.13202
  2. [2] Chr. BERG, Semi-groupes de Feller et le théorème de Hunt, Preprint Series no, 12, Matematisk Institut, University of Copenhagen, 1971. 
  3. [3] J. DENY, Méthodes hilbertiennes en théorie du potentiel, Potential theory (C.I.M.E., I ciclo, Stresa), Ed. Cremonese, Rome, 1970. Zbl0212.13401MR44 #1833
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  8. [8] R. GODEMENT, Sur la théorie des représentations unitaires, Ann. Math. 53 (1951), 68-124. Zbl0042.34606MR12,421d
  9. [9] R. GODEMENT, Introduction aux travaux de A. Selberg, Séminaire Bourbaki, Paris 1957, exposé 144. Zbl0202.40902
  10. [10] HARISH-CHANDRA, Spherical functions on a semi-simple Lie group II, Amer. J. Math. 80 (1958), 553-613. Zbl0093.12801MR21 #92
  11. [11] S. HELGASON, Differential Geometry and Symmetric spaces (Pure and Applied Mathematics 12), Academic Press, New York, London, 1962. Zbl0111.18101
  12. [12] B. KOSTANT, On the existence and irreducibility of certain series of representations, Bull. Amer. Math. Soc. 75 (1969), 627-642. Zbl0229.22026MR39 #7031
  13. [13] K. SATO, T. UENO, Multidimensional diffusion and the Markov process on the boundary, J. Math. Kyoto Univ. 4 (1965), 529-605. Zbl0219.60057MR33 #6702

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