Download PDF - Note on semigroups generated by positive Rockland operators on graded homogeneous groupsAll rights reserved. Copyright exceptions and limitations apply

ArticleOriginal scientific text

Title

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

Authors 1, 1, 1

Affiliations

  1. Mathematical Institute, University of Wrocław, Pl. Grunwaldzki 2/4, 50-384 Wrocław, Poland

Abstract

Let L be a positive Rockland operator of homogeneous degree d on a graded homogeneous group G and let pt 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 |p1(x)|≤Cexp(-cτ(x)d/(d-1)). Moreover, if G is not stratified, more precise estimates of p1 at infinity are given.

Bibliography

  1. [D] J. Dziubański, On semigroups generated by subelliptic operators on homogeneous groups, Colloq. Math. 64 (1993), 215-231.
  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.
  3. [FS] G. B. Folland and E. M. Stein, Hardy Spaces on Homogeneous Groups, Princeton Univ. Press, Princeton, 1982.
  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.
  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.
  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.
  8. [H] A. Hulanicki, Subalgebra of L1(G) associated with laplacian on a Lie group, Colloq. Math. 31 (1974), 259-287.
  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.
  10. [J] J. W. Jenkins, Dilations and gauges on nilpotent Lie groups, Colloq. Math. 41 (1979), 91-101.
  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.
Studia Mathematica
1994/110/2
Export to PBN, Opens in new tab
Pages:
115-126
Main language of publication
English
Received
1993-03-29
Accepted
1993-10-25
Published
1994
Exact and natural sciences