Note on semigroups generated by positive Rockland operators on graded homogeneous groups

Jacek Dziubański; Waldemar Hebisch; Jacek Zienkiewicz

Studia Mathematica (1994)

  • Volume: 110, Issue: 2, page 115-126
  • ISSN: 0039-3223

Abstract

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Let L be a positive Rockland operator of homogeneous degree d on a graded homogeneous group G and let p t be the convolution kernels of the semigroup generated by L. We prove that if τ(x) is a Riemannian distance of x from the unit element, then there are constants c>0 and C such that | p 1 ( x ) | ≤ C e x p ( - c τ ( x ) d / ( d - 1 ) ) . Moreover, if G is not stratified, more precise estimates of p 1 at infinity are given.

How to cite

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Dziubański, Jacek, Hebisch, Waldemar, and Zienkiewicz, Jacek. "Note on semigroups generated by positive Rockland operators on graded homogeneous groups." Studia Mathematica 110.2 (1994): 115-126. <http://eudml.org/doc/216104>.

@article{Dziubański1994,
abstract = {Let L be a positive Rockland operator of homogeneous degree d on a graded homogeneous group G and let $p_t$ be the convolution kernels of the semigroup generated by L. We prove that if τ(x) is a Riemannian distance of x from the unit element, then there are constants c>0 and C such that $|p_1(x)| ≤ Cexp(-cτ(x)^\{d/(d-1)\})$. Moreover, if G is not stratified, more precise estimates of $p_1$ at infinity are given.},
author = {Dziubański, Jacek, Hebisch, Waldemar, Zienkiewicz, Jacek},
journal = {Studia Mathematica},
keywords = {fundamental solution; upper bounds; kernel estimates; Rockland operator; homogeneous Lie group; convolution kernel; left-invariant differential operator},
language = {eng},
number = {2},
pages = {115-126},
title = {Note on semigroups generated by positive Rockland operators on graded homogeneous groups},
url = {http://eudml.org/doc/216104},
volume = {110},
year = {1994},
}

TY - JOUR
AU - Dziubański, Jacek
AU - Hebisch, Waldemar
AU - Zienkiewicz, Jacek
TI - Note on semigroups generated by positive Rockland operators on graded homogeneous groups
JO - Studia Mathematica
PY - 1994
VL - 110
IS - 2
SP - 115
EP - 126
AB - Let L be a positive Rockland operator of homogeneous degree d on a graded homogeneous group G and let $p_t$ be the convolution kernels of the semigroup generated by L. We prove that if τ(x) is a Riemannian distance of x from the unit element, then there are constants c>0 and C such that $|p_1(x)| ≤ Cexp(-cτ(x)^{d/(d-1)})$. Moreover, if G is not stratified, more precise estimates of $p_1$ at infinity are given.
LA - eng
KW - fundamental solution; upper bounds; kernel estimates; Rockland operator; homogeneous Lie group; convolution kernel; left-invariant differential operator
UR - http://eudml.org/doc/216104
ER -

References

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  1. [D] J. Dziubański, On semigroups generated by subelliptic operators on homogeneous groups, Colloq. Math. 64 (1993), 215-231. Zbl0837.43010
  2. [DH] J. Dziubański and A. Hulanicki, On semigroups generated by left-invariant positive differential operators on nilpotent Lie groups, Studia Math. 94 (1989), 81-95. Zbl0701.47020
  3. [FS] G. B. Folland and E. M. Stein, Hardy Spaces on Homogeneous Groups, Princeton Univ. Press, Princeton, 1982. Zbl0508.42025
  4. [He] W. Hebisch, Sharp pointwise estimates for the kernels of the semigroup generated by sums of even powers of vector fields on homogeneous groups, Studia Math. 95 (1989), 93-106. Zbl0693.22005
  5. [He1] W. Hebisch, Estimates on the semigroups generated by left invariant operators on Lie groups, J. Reine Angew. Math. 423 (1992), 1-45. 
  6. [HS] W. Hebisch and A. Sikora, A smooth subadditive homogeneous norm on a homogeneous group, Studia Math. 96 (1990), 231-236. Zbl0723.22007
  7. [HN] B. Helffer et J. Nourrigat, Caractérisation des opérateurs hypoelliptiques homogènes à gauche sur un groupe nilpotent gradué, Comm. Partial Differential Equations 4 (1979), 899-958. Zbl0423.35040
  8. [H] A. Hulanicki, Subalgebra of L 1 ( G ) associated with laplacian on a Lie group, Colloq. Math. 31 (1974), 259-287. Zbl0316.43005
  9. [HJ] A. Hulanicki and J. W. Jenkins, Nilpotent Lie groups and summability of eigenfunction expansions of Schrödinger operators, Studia Math. 80 (1984), 235-244. Zbl0564.43007
  10. [J] J. W. Jenkins, Dilations and gauges on nilpotent Lie groups, Colloq. Math. 41 (1979), 91-101. Zbl0434.22014
  11. [NRS] A. Nagel, F. Ricci and E. M. Stein, Harmonic analysis and fundamental solutions on nilpotent Lie groups, in: Analysis and Partial Differential Equations, Marcel Dekker, 1990, 249-275. 
  12. [P] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer, New York, 1983. Zbl0516.47023

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