Lipschitz continuity of densities of stable semigroups of measures

Paweł Głowacki

Colloquium Mathematicae (1993)

  • Volume: 66, Issue: 1, page 29-47
  • ISSN: 0010-1354

Abstract

top
In this paper we raise the question of regularity of the densities h t of a symmetric stable semigroup μ t of measures on the homogeneous group N under the mere assumption that the densities exist. (For a criterion of the existence of the densities of such semigroups see [11].)

How to cite

top

Głowacki, Paweł. "Lipschitz continuity of densities of stable semigroups of measures." Colloquium Mathematicae 66.1 (1993): 29-47. <http://eudml.org/doc/210232>.

@article{Głowacki1993,
abstract = {In this paper we raise the question of regularity of the densities $h_t$ of a symmetric stable semigroup $\{μ_t\}$ of measures on the homogeneous group N under the mere assumption that the densities exist. (For a criterion of the existence of the densities of such semigroups see [11].)},
author = {Głowacki, Paweł},
journal = {Colloquium Mathematicae},
keywords = {homogeneous group; dilations; accretive kernel; measures; Haar measure; convolution; holomorphic semigroup; orthogonal group; maximal function},
language = {eng},
number = {1},
pages = {29-47},
title = {Lipschitz continuity of densities of stable semigroups of measures},
url = {http://eudml.org/doc/210232},
volume = {66},
year = {1993},
}

TY - JOUR
AU - Głowacki, Paweł
TI - Lipschitz continuity of densities of stable semigroups of measures
JO - Colloquium Mathematicae
PY - 1993
VL - 66
IS - 1
SP - 29
EP - 47
AB - In this paper we raise the question of regularity of the densities $h_t$ of a symmetric stable semigroup ${μ_t}$ of measures on the homogeneous group N under the mere assumption that the densities exist. (For a criterion of the existence of the densities of such semigroups see [11].)
LA - eng
KW - homogeneous group; dilations; accretive kernel; measures; Haar measure; convolution; holomorphic semigroup; orthogonal group; maximal function
UR - http://eudml.org/doc/210232
ER -

References

top
  1. [1] A. P. Calderón and A. Zygmund, On singular integrals, Amer. J. Math. 78 (1956), 289-309. Zbl0072.11501
  2. [2] M. Christ, Hilbert transforms along curves. I. Nilpotent groups, Ann. of Math. 122 (1985), 575-596. Zbl0593.43011
  3. [3] L. Corwin and F. P. Greenleaf, Representations of Nilpotent Lie Groups and Their Applications. Part 1: Basic Theory and Examples, Cambridge University Press, Cambridge, 1990. Zbl0704.22007
  4. [4] M. Duflo, Représentations de semi-groupes de mesures sur un groupe localement compact, Ann. Inst. Fourier (Grenoble) 28 (3) (1978), 225-249. Zbl0368.22006
  5. [5] J. Dziubański and J. Zienkiewicz, Smoothness of densities of semigroups of measures on homogeneous groups, Colloq. Math., to appear. Zbl0838.43010
  6. [6] G. B. Folland, Subelliptic estimates and function spaces on nilpotent Lie groups, Ark. Mat. 13 (1975), 161-207. Zbl0312.35026
  7. [7] G. B. Folland, Lipschitz classes and Poisson integrals on stratified groups, Studia Math. 66 (1979), 37-55. Zbl0439.43005
  8. [8] G. B. Folland and E. M. Stein, Hardy Spaces on Homogeneous Groups, Princeton University Press, Princeton, 1982. Zbl0508.42025
  9. [9] P. Głowacki, Stable semigroups of measures on the Heisenberg group, Studia Math. 79 (1984), 105-138. Zbl0563.43002
  10. [10] P. Głowacki, Stable semi-groups of measures as commutative approximate identities on non- graded homogeneous groups, Invent. Math. 83 (1986), 557-582. Zbl0595.43006
  11. [11] P. Głowacki, The Rockland condition for nondifferential convolution operators, Duke Math. J. 58 (1989), 371-395. Zbl0678.43002
  12. [12] P. Głowacki and W. Hebisch, Pointwise estimates for the densities of stable semigroups of measures, Studia Math. 104 (1993), 243-258. Zbl0812.43005
  13. [13] P. Głowacki and A. Hulanicki, A semi-group of probability measures with non- smooth differentiable densities on a Lie group, Colloq. Math. 51 (1987), 131-139. Zbl0629.43001
  14. [14] B. Helffer et J. Nourrigat, Caractérisation des opérateurs hypoelliptiques homogènes invariants à gauche sur un groupe gradué, Comm. Partial Differential Equations 4 (8) (1979), 899-958. Zbl0423.35040
  15. [15] A. Hulanicki, A class of convolution semi-groups of measures on a Lie group, in: Lecture Notes in Math. 828, Springer, 1980, 82-101. Zbl0462.28009
  16. [16] G. Hunt, Semigroups of measures on Lie groups, Trans. Amer. Math. Soc. 81 (1956), 264-293. Zbl0073.12402
  17. [17] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer, New York, 1983. 
  18. [18] E. M. Stein, Boundary behavior of harmonic functions on symmetric spaces: maximal estimates for Poisson integrals, Invent. Math. 74 (1983), 63-83. Zbl0522.43007
  19. [19] E. M. Stein and G. Weiss, Introduction to Fourier Analysis on Euclidean Spaces, Princeton University Press, Princeton, 1975. Zbl0232.42007
  20. [20] K. Yosida, Functional Analysis, Springer, Berlin, 1980. 
  21. [21] F. Zo, A note on approximation of the identity, Studia Math. 55 (1976), 111-122. Zbl0326.44005

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.