Mathematical Analysis/Harmonic Analysis
Lower bounds for operators on graded Lie groups
Comptes Rendus. Mathématique, Volume 351 (2013) no. 1-2, pp. 13-18.

In this note we present a symbolic pseudo-differential calculus on any graded (nilpotent) Lie group and, as an application, a version of the sharp Gårding inequality. As a corollary, we obtain lower bounds for positive Rockland operators with variable coefficients as well as their Schwartz-hypoellipticity.

Dans cette note nous présentons un calcul pseudo-différentiel symbolique sur tous les groupes de Lie (nilpotents) gradués et, comme application, une version de lʼinégalité de Gårding. En découlent des bornes inférieures pour des opérateurs de Rockland positifs à coefficients variables ainsi que leur hypo-ellipticité Schwartz.

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Accepted:
Published online:
DOI: 10.1016/j.crma.2013.01.004
Fischer, Véronique 1; Ruzhansky, Michael 2

1 Universita degli studi di Padova, DMMMSA, Via Trieste 63, 35121 Padova, Italy
2 Department of Mathematics, Imperial College London, 180 Queenʼs Gate, London SW7 2AZ, United Kingdom
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Fischer, Véronique; Ruzhansky, Michael. Lower bounds for operators on graded Lie groups. Comptes Rendus. Mathématique, Volume 351 (2013) no. 1-2, pp. 13-18. doi : 10.1016/j.crma.2013.01.004. http://www.numdam.org/articles/10.1016/j.crma.2013.01.004/

[1] Bahouri, H.; Fermanian-Kammerer, C.; Gallagher, I. Phase-space analysis and pseudodifferential calculus on the Heisenberg group, Astérisque, Volume 342 (2012)

[2] Beals, R.; Greiner, P. Calculus on Heisenberg Manifolds, Princeton University Press, 1988

[3] Christ, M.; Geller, D.; Glowacki, P.; Polin, L. Pseudodifferential operators on groups with dilations, Duke Math. J., Volume 68 (1992), pp. 31-65

[4] Coifman, R.; Weiss, G. Analyse harmonique non-commutative sur certains espaces homogènes, Springer, 1971

[5] V. Fischer, M. Ruzhansky, Quantization on nilpotent Lie groups, monograph, in preparation.

[6] Folland, G.B. Subelliptic estimates and function spaces on nilpotent Lie groups, Ark. Mat., Volume 13 (1975), pp. 161-207

[7] Folland, G.B.; Stein, E. Hardy Spaces on Homogeneous Groups, Princeton University Press, 1982

[8] Helffer, B.; Nourrigat, J. Caracterisation des opérateurs hypoelliptiques homogènes invariants à gauche sur un groupe de Lie nilpotent gradué, Comm. Partial Differential Equations, Volume 4 (1979), pp. 899-958

[9] Ponge, R. Heisenberg calculus and spectral theory of hypoelliptic operators on Heisenberg manifolds, Mem. Amer. Math. Soc., Volume 194 (2008) (viii+134 pp)

[10] Ruzhansky, M.; Turunen, V. Pseudo-Differential Operators and Symmetries: Background Analysis and Advanced Topics, Birkhäuser, Basel, 2010

[11] Ruzhansky, M.; Turunen, V. Sharp Gårding inequality on compact Lie groups, J. Funct. Anal., Volume 260 (2011), pp. 2881-2901

[12] M. Ruzhansky, V. Turunen, Global quantization of pseudo-differential operators on compact Lie groups, SU(2) and 3-sphere, Int. Math. Res. Notices IMRN (2012) 58 pp., first published online April 17, 2012, . | DOI

[13] Taylor, M.E. Noncommutative microlocal analysis. I, Mem. Amer. Math. Soc., Volume 52 (1984) (iv+182 pp)

[14] Taylor, M.E. Noncommutative Harmonic Analysis, American Mathematical Society, 1986

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