Let be the Lie group endowed with the Riemannian symmetric space structure. Let be a distinguished basis of left-invariant vector fields of the Lie algebra of and define the Laplacian . In this paper we consider the first order Riesz transforms and , for . We prove that the operators , but not the , are bounded from the Hardy space to . We also show that the second-order Riesz transforms are bounded from to , while the transforms and , for , are not.
On considère le groupe de Lie muni de la structure Riemannienne d’espace symétrique. On choisit une base de champs vectoriels invariants à gauche de l’algèbre de Lie de et on définit le Laplacien . Dans cet article nous considérons les transformées de Riesz du premier ordre et , avec . Nous prouvons que les opérateurs , mais non pas les , sont bornés de l’espace de Hardy à . Nous démontrons aussi que les transformées de Riesz du deuxième ordre sont bornées de à , tandis que les transformées et , , ne sont pas bornées.
Keywords: Singular integrals, Riesz transforms, Hardy space, Lie groups, exponential growth
Mot clés : intégrales singulières, transformées de Riesz, espaces de Hardy, groupes de Lie, croissance exponentielle
Sjögren, Peter 1; Vallarino, Maria 2
@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}, pages = {1117--1151}, publisher = {Association des Annales de l{\textquoteright}institut Fourier}, volume = {58}, number = {4}, year = {2008}, doi = {10.5802/aif.2380}, mrnumber = {2427956}, language = {en}, url = {https://aif.centre-mersenne.org/articles/10.5802/aif.2380/} }
TY - JOUR AU - Sjögren, Peter AU - Vallarino, Maria TI - Boundedness from $H^1$ to $L^1$ of Riesz transforms on a Lie group of exponential growth JO - Annales de l'Institut Fourier PY - 2008 SP - 1117 EP - 1151 VL - 58 IS - 4 PB - Association des Annales de l’institut Fourier UR - https://aif.centre-mersenne.org/articles/10.5802/aif.2380/ DO - 10.5802/aif.2380 LA - en ID - AIF_2008__58_4_1117_0 ER -
%0 Journal Article %A Sjögren, Peter %A Vallarino, Maria %T Boundedness from $H^1$ to $L^1$ of Riesz transforms on a Lie group of exponential growth %J Annales de l'Institut Fourier %D 2008 %P 1117-1151 %V 58 %N 4 %I Association des Annales de l’institut Fourier %U https://aif.centre-mersenne.org/articles/10.5802/aif.2380/ %R 10.5802/aif.2380 %G en %F 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, Volume 58 (2008) no. 4, pp. 1117-1151. doi : 10.5802/aif.2380. https://aif.centre-mersenne.org/articles/10.5802/aif.2380/
[1] An application of homogenization theory to harmonic analysis: Harnack inequalities and Riesz transforms on Lie groups of polynomial growth, Can. J. Math., Volume 44 (1992), pp. 691-727 | DOI | MR | Zbl
[2] Sharp estimates for some functions of the Laplacian on noncompact symmetric spaces, Duke Math. J., Volume 65 (1992), pp. 257-297 | DOI | MR | Zbl
[3] Spherical Analysis on harmonic groups, Ann. Scuola Nom. Sup. Pisa Cl. Sci., Volume 23 (1996), pp. 643-679 | EuDML | Numdam | MR | Zbl
[4] Riesz transform on manifolds and Poincaré inequalities, Ann. Sci. École Norm. Sup., Volume 4 (2005), pp. 531-555 | EuDML | Numdam | MR | Zbl
[5] Riesz transform on manifolds and heat kernel regularity, Ann. Sci. École Norm. Sup., Volume 37 (2004), pp. 911-957 | EuDML | Numdam | MR | Zbl
[6] Étude des transformations de Riesz dans les variétés riemanniennes à courbure de Ricci minorée, Séminaire de Probabilité XXI, Lecture Notes in Math., Volume 1247 (1987), pp. 137-172 | DOI | EuDML | Numdam | MR | Zbl
[7] Duality of and BMO on positively curved manifolds and their characterizations, Lecture Notes in Math., Volume 1494 (1991), pp. 23-38 | DOI | MR | Zbl
[8] Extensions of Hardy spaces and their use in Analysis, Bull. Am. Math. Soc., Volume 83 (1977), pp. 569-645 | DOI | MR | Zbl
[9] Riesz transforms for , Trans. Amer. Math. Soc., Volume 351 (1999), pp. 1151-1169 | DOI | MR | Zbl
[10] Riesz transforms for , C. R. Acad. Sci. Paris Sér. I Math., Volume 332 (2001), pp. 975-980 | DOI | MR | Zbl
[11] Estimates of lower bounds for the heat kernel on conical manifolds and Riesz transform, Arch. Math. (Basel), Volume 83 (2004), pp. 229-242 | MR | Zbl
[12] Weak type estimates for heat kernel maximal functions on Lie groups, Trans. Amer. Math. Soc., Volume 323 (1991), pp. 637-649 | DOI | MR | Zbl
[13] Singular integrals associated to the Laplacian on the affine group , Ark. Mat., Volume 30 (1992), pp. 259-281 | DOI | MR | Zbl
[14] Singular integrals on Iwasawa groups of rank , J. Reine Angew. Math., Volume 479 (1996), pp. 39-66 | DOI | MR | Zbl
[15] Haar-like expansions and boundedness of a Riesz operator on a solvable Lie group, Math. Z., Volume 232 (1999), pp. 241-256 | DOI | MR | Zbl
[16] A note on maximal functions on a solvable Lie group, Arch. Math. (Basel), Volume 55 (1990), pp. 156-160 | MR | Zbl
[17] Multipliers and singular integrals on exponential growth groups, Math. Z., Volume 245 (2003), pp. 37-61 | DOI | MR | Zbl
[18] Remarques sur les transformées de Riesz sur les groupes de Lie nilpotents, C. R. Acad. Sci. Paris Sér. I Math., Volume 301 (1985), pp. 559-560 | MR | Zbl
[19] -boundedness of Riesz transforms and imaginary powers of the Laplacian on Riemannian manifolds, Ark. Mat., Volume 41 (2003), pp. 115-132 | DOI | MR | Zbl
[20] - boundedness of Riesz transforms on Riemannian manifolds and on graphs, Potential Anal., Volume 14 (2001), pp. 301-330 | DOI | MR | Zbl
[21] Analyse sur les groupes de Lie á croissance polynomiale, Ark. Mat., Volume 28 (1990), pp. 315-331 | DOI | MR | Zbl
[22] An estimate for a first-order Riesz operator on the affine group, Trans. Amer. Math. Soc., Volume 351 (1999), pp. 3301-3314 | DOI | MR | Zbl
[23] Harmonic Analysis, Princeton University Press, 1993 | MR | Zbl
[24] Spaces and on exponential growth groups (2007) (submitted)
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