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## Colloquium Mathematicum

1994 | 67 | 2 | 197-216
Tytuł artykułu

### On positive Rockland operators

Autorzy
Treść / Zawartość
Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
Let G be a homogeneous Lie group with a left Haar measure dg and L the action of G as left translations on $L_p(G;dg)$. Further, let H = dL(C) denote a homogeneous operator associated with L. If H is positive and hypoelliptic on $L_2$ we prove that it is closed on each of the $L_p$-spaces, p ∈ 〈 1,∞〉, and that it generates a semigroup S with a smooth kernel K which, with its derivatives, satisfies Gaussian bounds. The semigroup is holomorphic in the open right half-plane on all the $L_p$-spaces, p ∈ [1,∞]. Further extensions of these results to nonhomogeneous operators and general representations are also given.
Słowa kluczowe
Czasopismo
Rocznik
Tom
Numer
Strony
197-216
Opis fizyczny
Daty
wydano
1994
otrzymano
1993-09-14
Twórcy
autor
• IRMAR, Université de Rennes I, Campus de Beaulieu, 35042 Rennes Cedex, France
autor
• Department of Mathematics, and Computing Science, Eindhoven University of Technology, P.O. Box 513, 5600 MB Eindhoven, The Netherlands
autor
• Centre for Mathematics and its Applications School of Mathematical Sciences Australian National University Canberra, ACT 0200 Australia
Bibliografia
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• [AMT] P. Auscher, A. McIntosh and Ph. Tchamitchian, Noyau de la chaleur d'opérateurs elliptiques complexes, Math. Research Letters 1 (1994), 37-45.
• [BrR1] O. Bratteli and D. W. Robinson, Operator Algebras and Quantum Statistical Mechanics, Vol. 1, 2nd ed., Springer, New York, 1987.
• [BrR2] O. Bratteli and D. W. Robinson, Subelliptic operators on Lie groups: variable coefficients, Acta Appl. Math. (1994), to appear.
• [BER] R. J. Burns, A. F. M. ter Elst and D. W. Robinson, $L_p$-regularity of subelliptic operators on Lie groups, J. Operator Theory 30 (1993), to appear.
• [Dzi] J. Dziubański, On semigroups generated by subelliptic operators on homogeneous groups, Colloq. Math. 64 (1993), 215-231.
• [DHZ] J. Dziubański, W. Hebisch and J. Zienkiewicz, Note on semigroups generated by positive Rockland operators on graded homogeneous groups, Studia Math. 110 (1994), 115-126.
• [ElR1] A. F. M. ter Elst and D. W. Robinson, Subcoercivity and subelliptic operators on Lie groups II: The general case, Potential Anal. (1994), to appear.
• [ElR2] A. F. M. ter Elst and D. W. Robinson, Subcoercive and subelliptic operators on Lie groups: variable coefficients, Publ. RIMS Kyoto Univ. 29 (1993), 745-801.
• [ElR3] A. F. M. ter Elst and D. W. Robinson, Functional analysis of subelliptic operators on Lie groups, J. Operator Theory 30 (1993), to appear.
• [ElR4] A. F. M. ter Elst and D. W. Robinson, Weighted strongly elliptic operators on Lie groups, J. Funct. Anal. (1994), to appear.
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• [HeS] W. Hebisch and A. Sikora, A smooth subadditive homogeneous norm on a homogeneous group, ibid. 96 (1990), 231-236.
• [HeN1] B. Helffer et J. Nourrigat, Caractérisation des opérateurs hypoelliptiques homogènes invariants à gauche sur un groupe de Lie nilpotent gradué, Comm. Partial Differential Equations 4 (1979), 899-958.
• [HeN2] B. Helffer et J. Nourrigat, Hypoellipticité maximale pour des opérateurs polynômes de champs de vecteurs, Progr. Math. 58, Birkhäuser, Boston, 1985.
• [Kat] T. Kato, Perturbation Theory for Linear Operators, 2nd ed., Grundlehren Math. Wiss. 132, Springer, Berlin, 1984.
• [Mil] K. G. Miller, Parametrices for hypoelliptic operators on step two nilpotent Lie groups, Comm. Partial Differential Equations 5 (1980), 1153-1184.
• [NeS] E. Nelson and W. F. Stinespring, Representation of elliptic operators in an enveloping algebra, Amer. J. Math. 81 (1959), 547-560.
• [Nir] L. Nirenberg, Remarks on strongly elliptic partial differential operators, Comm. Pure Appl. Math. 8 (1955), 649-675.
• [Rob] D. W. Robinson, Elliptic Operators and Lie Groups, Oxford Math. Monographs, Oxford University Press, Oxford, 1991.
• [Roc] C. Rockland, Hypoellipticity for the Heisenberg group, Trans. Amer. Math. Soc. 240 (1978), 1-52.
• [VSC] N. T. Varopoulos, L. Saloff-Coste and T. Coulhon, Analysis and Geometry on Groups, Cambridge Tracts in Math. 100, Cambridge University Press, Cambridge, 1992.
Typ dokumentu
Bibliografia
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