Boundedness from $H^1$ to $L^1$ of Riesz transforms on a Lie group of exponential growth

Sjögren, Peter; Vallarino, Maria

Boundedness from

H^{1}

L^{1}

of Riesz transforms on a Lie group of exponential growth
[Transformées de Riesz bornées de

H^{1}

L^{1}

sur un groupe de Lie à croissance exponentielle]

Sjögren, Peter ; Vallarino, Maria

Annales de l'Institut Fourier, Tome 58 (2008), p. 1117-1151 / Harvested from Numdam

Access to full text
Full (PDF)
Full (DJVU)

Résumé
Abstract

On considère le groupe de Lie $G = ℝ^{2} ⋉ ℝ^{+}$ muni de la structure Riemannienne d’espace symétrique. On choisit une base $X_{0},$ $X_{1},$ $X_{2}$ de champs vectoriels invariants à gauche de l’algèbre de Lie de $G$ et on définit le Laplacien $Δ = - (X_{0}^{2} + X_{1}^{2} + X_{2}^{2})$ . Dans cet article nous considérons les transformées de Riesz du premier ordre $R_{i} = X_{i} Δ^{- 1 / 2}$ et $S_{i} = Δ^{- 1 / 2} X_{i}$ , avec $i = 0, 1, 2$ . Nous prouvons que les opérateurs $R_{i}$ , mais non pas les $S_{i}$ , sont bornés de l’espace de Hardy $H^{1}$ à $L^{1}$ . Nous démontrons aussi que les transformées de Riesz du deuxième ordre $T_{i j} = X_{i} Δ^{- 1} X_{j}$ sont bornées de $H^{1}$ à $L^{1}$ , tandis que les transformées $S_{i j} = Δ^{- 1} X_{i} X_{j}$ et $R_{i j} = X_{i} X_{j} Δ^{- 1}$ , $i, j = 0, 1, 2$ , ne sont pas bornées.

Let $G$ be the Lie group $ℝ^{2} ⋉ ℝ^{+}$ endowed with the Riemannian symmetric space structure. Let $X_{0}, X_{1}, X_{2}$ be a distinguished basis of left-invariant vector fields of the Lie algebra of $G$ and define the Laplacian $Δ = - (X_{0}^{2} + X_{1}^{2} + X_{2}^{2})$ . In this paper we consider the first order Riesz transforms $R_{i} = X_{i} Δ^{- 1 / 2}$ and $S_{i} = Δ^{- 1 / 2} X_{i}$ , for $i = 0, 1, 2$ . We prove that the operators $R_{i}$ , but not the $S_{i}$ , are bounded from the Hardy space $H^{1}$ to $L^{1}$ . We also show that the second-order Riesz transforms $T_{i j} = X_{i} Δ^{- 1} X_{j}$ are bounded from $H^{1}$ to $L^{1}$ , while the transforms $S_{i j} = Δ^{- 1} X_{i} X_{j}$ and $R_{i j} = X_{i} X_{j} Δ^{- 1}$ , for $i, j = 0, 1, 2$ , are not.

Publié le : 2008-01-01

MR 2427956 | Zbl pre05303671

DOI : https://doi.org/10.5802/aif.2380
Classification: 43A80, 42B20, 42B30, 22E30
Mots clés: intégrales singulières, transformées de Riesz, espaces de Hardy, groupes de Lie, croissance exponentielle

@article{AIF_2008__58_4_1117_0,
     author = {Sj\"ogren, Peter and Vallarino, Maria},
     title = {Boundedness from $H^1$ to $L^1$ of Riesz transforms on a Lie group of exponential growth},
     journal = {Annales de l'Institut Fourier},
     volume = {58},
     year = {2008},
     pages = {1117-1151},
     doi = {10.5802/aif.2380},
     zbl = {pre05303671},
     mrnumber = {2427956},
     language = {en},
     url = {http://dml.mathdoc.fr/item/AIF_2008__58_4_1117_0}
}

Sjögren, Peter; Vallarino, Maria. Boundedness from $H^1$ to $L^1$ of Riesz transforms on a Lie group of exponential growth. Annales de l'Institut Fourier, Tome 58 (2008) pp. 1117-1151. doi : 10.5802/aif.2380. http://gdmltest.u-ga.fr/item/AIF_2008__58_4_1117_0/

Bibliographie

[1] Alexopoulos, G. An application of homogenization theory to harmonic analysis: Harnack inequalities and Riesz transforms on Lie groups of polynomial growth, Can. J. Math., Tome 44 (1992), pp. 691-727 | Article | MR 1178564 | Zbl 0792.22005

[2] Anker, J.P. Sharp estimates for some functions of the Laplacian on noncompact symmetric spaces, Duke Math. J., Tome 65 (1992), pp. 257-297 | Article | MR 1150587 | Zbl 0764.43005

[3] Anker, J.P.; Damek, E.; Yacoub, C. Spherical Analysis on harmonic $A N$ groups, Ann. Scuola Nom. Sup. Pisa Cl. Sci., Tome 23 (1996), pp. 643-679 | Numdam | MR 1469569 | Zbl 0881.22008

[4] Auscher, P.; Coulhon, T. Riesz transform on manifolds and Poincaré inequalities, Ann. Sci. École Norm. Sup., Tome 4 (2005), pp. 531-555 | Numdam | MR 2185868 | Zbl 1116.58023

[5] Auscher, P.; Coulhon, T.; Duong, X.T.; Hofmann, S. Riesz transform on manifolds and heat kernel regularity, Ann. Sci. École Norm. Sup., Tome 37 (2004), pp. 911-957 | Numdam | MR 2119242 | Zbl 1086.58013

[6] Bakry, D. Étude des transformations de Riesz dans les variétés riemanniennes à courbure de Ricci minorée, Séminaire de Probabilité XXI, Lecture Notes in Math., Tome 1247 (1987), pp. 137-172 | Article | Numdam | MR 941980 | Zbl 0629.58018

[7] Chen, J.Ch.; Luo, Ch. Duality of $H^{1}$ and BMO on positively curved manifolds and their characterizations, Lecture Notes in Math., Tome 1494 (1991), pp. 23-38 | Article | MR 1187062 | Zbl 0812.43001

[8] Coifman, R.R.; Weiss, G. Extensions of Hardy spaces and their use in Analysis, Bull. Am. Math. Soc., Tome 83 (1977), pp. 569-645 | Article | MR 447954 | Zbl 0358.30023

[9] Coulhon, T.; Duong, X.T. Riesz transforms for $1 \leq p \leq 2$ , Trans. Amer. Math. Soc., Tome 351 (1999), pp. 1151-1169 | Article | MR 1458299 | Zbl 0973.58018

[10] Coulhon, T.; Duong, X.T. Riesz transforms for $p > 2$ , C. R. Acad. Sci. Paris Sér. I Math., Tome 332 (2001), pp. 975-980 | Article | MR 1838122 | Zbl 0987.43001

[11] Coulhon, T.; Duong, X.T. Estimates of lower bounds for the heat kernel on conical manifolds and Riesz transform, Arch. Math. (Basel), Tome 83 (2004), pp. 229-242 | MR 2108551 | Zbl 1076.58017

[12] Cowling, M.; Gaudry, G.; Giulini, S.; Mauceri, G. Weak type $(1, 1)$ estimates for heat kernel maximal functions on Lie groups, Trans. Amer. Math. Soc., Tome 323 (1991), pp. 637-649 | Article | MR 967310 | Zbl 0722.22006

[13] Gaudry, G.; Qian, T.; Sjögren, P. Singular integrals associated to the Laplacian on the affine group $a x + b$ , Ark. Mat., Tome 30 (1992), pp. 259-281 | Article | MR 1289755 | Zbl 0776.43003

[14] Gaudry, G.; Sjögren, P. Singular integrals on Iwasawa $N A$ groups of rank $1$ , J. Reine Angew. Math., Tome 479 (1996), pp. 39-66 | Article | MR 1414387 | Zbl 0855.43010

[15] Gaudry, G.; Sjögren, P. Haar-like expansions and boundedness of a Riesz operator on a solvable Lie group, Math. Z., Tome 232 (1999), pp. 241-256 | Article | MR 1718685 | Zbl 0936.43006

[16] Giulini, S.; Sjögren, P. A note on maximal functions on a solvable Lie group, Arch. Math. (Basel), Tome 55 (1990), pp. 156-160 | MR 1064383 | Zbl 0693.42020

[17] Hebisch, W.; Steger, T. Multipliers and singular integrals on exponential growth groups, Math. Z., Tome 245 (2003), pp. 37-61 | Article | MR 2023952 | Zbl 1035.43001

[18] Lohoué, N.; Varopoulos, N. Remarques sur les transformées de Riesz sur les groupes de Lie nilpotents, C. R. Acad. Sci. Paris Sér. I Math., Tome 301 (1985), p. 559-560 | MR 816628 | Zbl 0582.43003

[19] Marias, M.; Russ, E. $H^{1}$ -boundedness of Riesz transforms and imaginary powers of the Laplacian on Riemannian manifolds, Ark. Mat., Tome 41 (2003), pp. 115-132 | Article | MR 1971944 | Zbl 1038.42016

[20] Russ, E. $H^{1}$ - $L^{1}$ boundedness of Riesz transforms on Riemannian manifolds and on graphs, Potential Anal., Tome 14 (2001), pp. 301-330 | Article | MR 1822920 | Zbl 0982.42008

[21] Saloff-Coste, L. Analyse sur les groupes de Lie á croissance polynomiale, Ark. Mat., Tome 28 (1990), pp. 315-331 | Article | MR 1084020 | Zbl 0715.43009

[22] Sjögren, P. An estimate for a first-order Riesz operator on the affine group, Trans. Amer. Math. Soc., Tome 351 (1999), pp. 3301-3314 | Article | MR 1475695 | Zbl 0920.43006

[23] Stein, E.M. Harmonic Analysis, Princeton University Press (1993) | MR 1232192 | Zbl 0821.42001

[24] Vallarino, M. Spaces $H^{1}$ and $B M O$ on exponential growth groups (2007) (submitted)