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1997 | 74 | 2 | 191-218
Tytuł artykułu

Heat kernels and Riesz transforms on nilpotent Lie groups

Treść / Zawartość
Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
We consider pure mth order subcoercive operators with complex coefficients acting on a connected nilpotent Lie group. We derive Gaussian bounds with the correct small time singularity and the optimal large time asymptotic behaviour on the heat kernel and all its derivatives, both right and left. Further we prove that the Riesz transforms of all orders are bounded on the Lp -spaces with p ∈ (1, ∞). Finally, for second-order operators with real coefficients we derive matching Gaussian lower bounds and deduce Harnack inequalities valid for all times.
Słowa kluczowe
Rocznik
Tom
74
Numer
2
Strony
191-218
Opis fizyczny
Daty
wydano
1998
otrzymano
1996-12-03
Twórcy
  • Department of Mathematics and Computing Science, Eindhoven University of Technology, P.O. Box 513, 5600 MB Eindhoven, The Netherlands
  • Centre for Mathematics and its Applications, School of Mathematical Sciences, Australian National University, Canberra, ACT 0200, Australia
autor
  • Centre for Mathematics and its Applications, School of Mathematical Sciences, Australian National University, Canberra, ACT 0200, Australia
Bibliografia
  • [ADM] D. Albrecht, X. Duong and A. McIntosh, Operator theory and harmonic analysis, in: Instructional Workshop on Analysis and Geometry, Part III, Proc. Centre Math. Appl. Austral. Nat. Univ. 34, Austral. Nat. Univ., Canberra, 1996, 77-136.
  • [Ale] G. Alexopoulos, An application of homogenization theory to harmonic analysis: Harnack inequalities and Riesz transforms on Lie groups of polynomial growth, Canad. J. Math. 44 (1992), 691-727.
  • [BeL] J. Bergh and J. Löfström, Interpolation Spaces. An Introduction, Grundlehren Math. Wiss. 223, Springer, Berlin, 1976.
  • [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 31 (1994), 165-187.
  • [BuB] P. L. Butzer and H. Berens, Semi-groups of Operators and Approximation, Grundlehren Math. Wiss. 145, Springer, Berlin, 1967.
  • [CoW1] R. R. Coifman et G. Weiss, Analyse harmonique noncommutative sur certains espaces homogènes, Lecture Notes in Math. 242, Springer, Berlin, 1971.
  • [CoW2] R. R. Coifman et G. Weiss, Transference Methods in Analysis, CBMS Regional Conf. Ser. in Math. 31, Amer. Math. Soc., Providence, 1977.
  • [CDMY] M. Cowling, I. Doust, A. McIntosh and A. Yagi, Banach space operators with a bounded H^∞ functional calculus, J. Austral. Math. Soc. Ser. A 60 (1996), 51-89.
  • [DuR] X. T. Duong and D. W. Robinson, Semigroup kernels, Poisson bounds and holomorphic functional calculus, J. Funct. Anal. 141 (1996), 89-129.
  • [ElR1] A. F. M. ter Elst and D. W. Robinson, Subelliptic operators on Lie groups: regularity, J. Austral. Math. Soc. Ser. A 57 (1994), 179-229.
  • [ElR2] A. F. M. ter Elst and D. W. Robinson, Functional analysis of subelliptic operators on Lie groups, J. Operator Theory 31 (1994), 277-301.
  • [ElR3] A. F. M. ter Elst and D. W. Robinson, Subcoercivity and subelliptic operators on Lie groups I: Free nilpotent groups, Potential Anal. 3 (1994), 283-337.
  • [ElR4] A. F. M. ter Elst and D. W. Robinson, Subcoercivity and subelliptic operators on Lie groups II: The general case, ibid. 4 (1995), 205-243.
  • [ElR5] A. F. M. ter Elst and D. W. Robinson, Reduced heat kernels on nilpotent Lie groups, Comm. Math. Phys. 173 (1995), 475-511.
  • [ElR6] A. F. M. ter Elst and D. W. Robinson, Elliptic operators on Lie groups, Acta Appl. Math. 44 (1996), 133-150.
  • [Fol] G. B. Folland, Subelliptic estimates and function spaces on nilpotent Lie groups, Ark. Mat. 13 (1975), 161-207.
  • [GQS] G. I. Gaudry, T. Qian and P. Sjögren, Singular integrals associated to the Laplacian on the affine group ax + b, ibid. 30 (1992), 259-281.
  • [HeR] E. Hewitt and K. A. Ross, Abstract Harmonic Analysis II, Grundlehren Math. Wiss. 152, Springer, Berlin, 1970.
  • [LoV] N. Lohoué et N. T. Varopoulos, Remarques sur les transformées de Riesz sur les groupes de Lie nilpotents, C. R. Acad. Sci. Paris Sér. I 301 (1985), 559-560.
  • [McI] A. McIntosh, Operators which have an H_∞ functional calculus, in: B. Jefferies, A. McIntosh, and W. J. Ricker (eds.), Miniconference on Operator Theory and Partial Differential Equations, Proc. Centre Math. Anal. Austral. Nat. Univ. 14, Austral. Nat. Univ., Canberra, 1986, 210-231.
  • [Pat] A. L. T. Paterson, Amenability, Math. Surveys Monographs 29, Amer. Math. Soc., Providence, 1988.
  • [Rob] D. W. Robinson, Elliptic Operators and Lie Groups, Oxford Math. Monographs, Oxford Univ. Press, New York, 1991.
  • [Sal] L. Saloff-Coste, Analyse sur les groupes de Lie à croissance polynômiale, Ark. Mat. 28 (1990), 315-331.
  • [Ste] E. M. Stein, Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals, Princeton Univ. Press, Princeton, 1993.
  • [Var] V. S. Varadarajan, Lie Groups, Lie Algebras, and their Representations, Grad. Texts in Math. 102, Springer, New York, 1984.
Typ dokumentu
Bibliografia
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Identyfikator YADDA
bwmeta1.element.bwnjournal-article-cmv74i2p191bwm
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