Heat kernels and Riesz transforms on nilpotent Lie groups

A. ter Elst; Derek Robinson; Adam Sikora

Colloquium Mathematicae (1998)

  • Volume: 74, Issue: 2, page 191-218
  • ISSN: 0010-1354

Abstract

top
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.

How to cite

top

ter Elst, A., Robinson, Derek, and Sikora, Adam. "Heat kernels and Riesz transforms on nilpotent Lie groups." Colloquium Mathematicae 74.2 (1998): 191-218. <http://eudml.org/doc/210510>.

@article{terElst1998,
abstract = {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.},
author = {ter Elst, A., Robinson, Derek, Sikora, Adam},
journal = {Colloquium Mathematicae},
keywords = {subellipticity; comparison operator; Gaussian bounds; Riesz operators; Lipschitz spaces; Harnack-type inequalities},
language = {eng},
number = {2},
pages = {191-218},
title = {Heat kernels and Riesz transforms on nilpotent Lie groups},
url = {http://eudml.org/doc/210510},
volume = {74},
year = {1998},
}

TY - JOUR
AU - ter Elst, A.
AU - Robinson, Derek
AU - Sikora, Adam
TI - Heat kernels and Riesz transforms on nilpotent Lie groups
JO - Colloquium Mathematicae
PY - 1998
VL - 74
IS - 2
SP - 191
EP - 218
AB - 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.
LA - eng
KW - subellipticity; comparison operator; Gaussian bounds; Riesz operators; Lipschitz spaces; Harnack-type inequalities
UR - http://eudml.org/doc/210510
ER -

References

top
  1. [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. Zbl0903.47010
  2. [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. Zbl0792.22005
  3. [BeL] J. Bergh and J. Löfström, Interpolation Spaces. An Introduction, Grundlehren Math. Wiss. 223, Springer, Berlin, 1976. Zbl0344.46071
  4. [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. Zbl0857.58044
  5. [BuB] P. L. Butzer and H. Berens, Semi-groups of Operators and Approximation, Grundlehren Math. Wiss. 145, Springer, Berlin, 1967. 
  6. [CoW1] R. R. Coifman et G. Weiss, Analyse harmonique noncommutative sur certains espaces homogènes, Lecture Notes in Math. 242, Springer, Berlin, 1971. 
  7. [CoW2] R. R. Coifman et G. Weiss, Transference Methods in Analysis, CBMS Regional Conf. Ser. in Math. 31, Amer. Math. Soc., Providence, 1977. 
  8. [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. Zbl0853.47010
  9. [DuR] X. T. Duong and D. W. Robinson, Semigroup kernels, Poisson bounds and holomorphic functional calculus, J. Funct. Anal. 141 (1996), 89-129. Zbl0932.47013
  10. [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. Zbl0832.43008
  11. [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. Zbl0851.43005
  12. [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. Zbl0923.22005
  13. [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. Zbl0841.43016
  14. [ElR5] A. F. M. ter Elst and D. W. Robinson, Reduced heat kernels on nilpotent Lie groups, Comm. Math. Phys. 173 (1995), 475-511. Zbl0838.22002
  15. [ElR6] A. F. M. ter Elst and D. W. Robinson, Elliptic operators on Lie groups, Acta Appl. Math. 44 (1996), 133-150. Zbl0856.22015
  16. [Fol] G. B. Folland, Subelliptic estimates and function spaces on nilpotent Lie groups, Ark. Mat. 13 (1975), 161-207. Zbl0312.35026
  17. [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. Zbl0776.43003
  18. [HeR] E. Hewitt and K. A. Ross, Abstract Harmonic Analysis II, Grundlehren Math. Wiss. 152, Springer, Berlin, 1970. 
  19. [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. Zbl0582.43003
  20. [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. 
  21. [Pat] A. L. T. Paterson, Amenability, Math. Surveys Monographs 29, Amer. Math. Soc., Providence, 1988. 
  22. [Rob] D. W. Robinson, Elliptic Operators and Lie Groups, Oxford Math. Monographs, Oxford Univ. Press, New York, 1991. 
  23. [Sal] L. Saloff-Coste, Analyse sur les groupes de Lie à croissance polynômiale, Ark. Mat. 28 (1990), 315-331. Zbl0715.43009
  24. [Ste] E. M. Stein, Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals, Princeton Univ. Press, Princeton, 1993. 
  25. [Var] V. S. Varadarajan, Lie Groups, Lie Algebras, and their Representations, Grad. Texts in Math. 102, Springer, New York, 1984. 

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.