Asymptotics of sums of subcoercive operators

Nick Dungey; A. ter Elst; Derek Robinson

Colloquium Mathematicae (1999)

  • Volume: 82, Issue: 2, page 231-260
  • ISSN: 0010-1354

Abstract

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We examine the asymptotic, or large-time, behaviour of the semigroup kernel associated with a finite sum of homogeneous subcoercive operators acting on a connected Lie group of polynomial growth. If the group is nilpotent we prove that the kernel is bounded by a convolution of two Gaussians whose orders correspond to the highest and lowest orders of the homogeneous subcoercive components of the generator. Moreover we establish precise asymptotic estimates on the difference of the kernel and the kernel corresponding to the lowest order homogeneous component. We also prove boundedness of a range of Riesz transforms with the range again determined by the highest and lowest orders. Finally we analyze similar properties on general groups of polynomial growth and establish positive results for local direct products of compact and nilpotent groups.

How to cite

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Dungey, Nick, ter Elst, A., and Robinson, Derek. "Asymptotics of sums of subcoercive operators." Colloquium Mathematicae 82.2 (1999): 231-260. <http://eudml.org/doc/210760>.

@article{Dungey1999,
abstract = {We examine the asymptotic, or large-time, behaviour of the semigroup kernel associated with a finite sum of homogeneous subcoercive operators acting on a connected Lie group of polynomial growth. If the group is nilpotent we prove that the kernel is bounded by a convolution of two Gaussians whose orders correspond to the highest and lowest orders of the homogeneous subcoercive components of the generator. Moreover we establish precise asymptotic estimates on the difference of the kernel and the kernel corresponding to the lowest order homogeneous component. We also prove boundedness of a range of Riesz transforms with the range again determined by the highest and lowest orders. Finally we analyze similar properties on general groups of polynomial growth and establish positive results for local direct products of compact and nilpotent groups.},
author = {Dungey, Nick, ter Elst, A., Robinson, Derek},
journal = {Colloquium Mathematicae},
keywords = {subcoercive operators; connected Lie group; Riesz transforms; nilpotent Lie groups},
language = {eng},
number = {2},
pages = {231-260},
title = {Asymptotics of sums of subcoercive operators},
url = {http://eudml.org/doc/210760},
volume = {82},
year = {1999},
}

TY - JOUR
AU - Dungey, Nick
AU - ter Elst, A.
AU - Robinson, Derek
TI - Asymptotics of sums of subcoercive operators
JO - Colloquium Mathematicae
PY - 1999
VL - 82
IS - 2
SP - 231
EP - 260
AB - We examine the asymptotic, or large-time, behaviour of the semigroup kernel associated with a finite sum of homogeneous subcoercive operators acting on a connected Lie group of polynomial growth. If the group is nilpotent we prove that the kernel is bounded by a convolution of two Gaussians whose orders correspond to the highest and lowest orders of the homogeneous subcoercive components of the generator. Moreover we establish precise asymptotic estimates on the difference of the kernel and the kernel corresponding to the lowest order homogeneous component. We also prove boundedness of a range of Riesz transforms with the range again determined by the highest and lowest orders. Finally we analyze similar properties on general groups of polynomial growth and establish positive results for local direct products of compact and nilpotent groups.
LA - eng
KW - subcoercive operators; connected Lie group; Riesz transforms; nilpotent Lie groups
UR - http://eudml.org/doc/210760
ER -

References

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  1. [ADM] Albrecht, D., Duong, X., and McIntosh, A., Operator theory and harmonic analysis, in: Instructional Workshop on Analysis and Geometry, Part III, Proc. Centre Math. Appl. 34, Australian National Univ., Canberra, 1996, 77-136. 
  2. [BaD] Barbatis, G., and Davies, E. B., Sharp bounds on heat kernels of higher order uniformly elliptic operators, J. Operator Theory 36 (1996), 179-198. Zbl0869.35048
  3. [Dav] Davies, E. B., One-Parameter Semigroups, London Math. Soc. Monographs 15, Academic Press, London, 1980. 
  4. [Dun] Dungey, N., Higher order operators and Gaussian bounds on Lie groups of polynomial growth, Research Report MRR 053-98, Australian National Univ., Canberra, 1998. 
  5. [DERS] Dungey, N., Elst, A. F. M. ter, Robinson, D. W., and Sikora, A., Asymptotics of subcoercive semigroups on nilpotent Lie groups, J. Operator Theory (1999), to appear. 
  6. [ElR1] Elst, A. F. M. ter, and Robinson, D. W., Subcoercivity and subelliptic operators on Lie groups I: Free nilpotent groups, Potential Anal. 3 (1994), 283-337. Zbl0923.22005
  7. [ElR2] Elst, A. F. M. ter, and Robinson, D. W., Weighted strongly elliptic operators on Lie groups, J. Funct. Anal. 125 (1994), 548-603. Zbl0815.58023
  8. [ElR3] Elst, A. F. M. ter, and Robinson, D. W., Subcoercivity and subelliptic operators on Lie groups II: The general case, Potential Anal. 4 (1995), 205-243. Zbl0841.43016
  9. [ElR4] Elst, A. F. M. ter, and Robinson, D. W., Weighted subcoercive operators on Lie groups, J. Funct. Anal. 157 (1998), 88-163. Zbl0910.22005
  10. [ElR5] Elst, A. F. M. ter, and Robinson, D. W., Local lower bounds on heat kernels, Positivity 2 (1998), 123-151. Zbl0916.47025
  11. [ERS1] Elst, A. F. M. ter, Robinson, D. W., and Sikora, A., Heat kernels and Riesz transforms on nilpotent Lie groups, Colloq. Math. 74 (1997), 191-218. Zbl0891.35030
  12. [ERS2] Elst, A. F. M. ter, Robinson, D. W., and Sikora, A., Riesz transforms and Lie groups of polynomial growth, J. Funct. Anal. 162 (1999), 14-51. Zbl0953.22011
  13. [NRS] Nagel, A., Ricci, F., and Stein, E. M., Harmonic analysis and fundamental solutions on nilpotent Lie groups, in: C. Sadosky (ed.), Analysis and Partial Differential Equations, Lecture Notes in Pure and Appl. Math. 122, Dekker, New York, 1990, 249-275. 
  14. [Rob] Robinson, D. W., Elliptic Operators and Lie Groups, Oxford Math. Monographs, Oxford Univ. Press, Oxford, 1991. 
  15. [VSC] Varopoulos, N. T., Saloff-Coste, L., and Coulhon, T., Analysis and Geometry on Groups, Cambridge Tracts in Math. 100, Cambridge Univ. Press, Cambridge, 1992. 

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