Download PDF - Estimates for the Poisson kernels and their derivatives on rank one NA groupsAll rights reserved. Copyright exceptions and limitations apply

ArticleOriginal scientific text

Title

Estimates for the Poisson kernels and their derivatives on rank one NA groups

Authors 1, 1, 2, 1

Affiliations

  1. Institute of Mathematics, The University of Wrocław, Plac Grunwaldzki 2/4, 50-384 Wrocław, Poland
  2. hulanick@math.uni.wroc.pl

Abstract

For rank one solvable Lie groups of the type NA estimates for the Poisson kernels and their derivatives are obtained. The results give estimates on the Poisson kernel and its derivatives in a natural parametrization of the Poisson boundary (minus one point) of a general homogeneous, simply connected manifold of negative curvature.

Bibliography

  1. [A] A. Ancona, Negatively curved manifolds, elliptic operators, and the Martin boundary, Ann. of Math. 125 (1987), 495-535.
  2. [B] M. Babillot, Théorie de renouvellement pour des chaînes semi-markoviennes transientes, Ann. Inst. H. Poincaré 24 (1988), 507-569.
  3. [BBE] M. Babillot, P. Bougerol and L. Elie, The random difference equation Xn=AnXn-1+Bn in the critical case, Ann. Probab. 25 (1997), 478-493.
  4. [BR] L. Birgé et A. Raugi, Fonctions harmoniques sur les groupes moyennables, C. R. Acad. Sci. Paris 278 (1974), 1287-1289.
  5. [D1] E. Damek, Left-invariant degenerate elliptic operators on semidirect extensions of homogeneous groups, Studia Math. 89 (1988), 169-195.
  6. [D2] E. Damek, Pointwise estimates for the Poisson kernel on NA groups by the Ancona method, Ann. Fac. Sci. Toulouse 5 (1996), 421-441.
  7. [D3] E. Damek, Fundamental solutions of differential operators on homogeneous manifolds of negative curvature and related Riesz transforms, Colloq. Math. 73 (1997), 229-249.
  8. [DH1] E. Damek and A. Hulanicki, Boundaries for left-invariant subelliptic operators on semidirect products of nilpotent and abelian groups, J. Reine Angew. Math. 411 (1990), 1-38.
  9. [DH2] E. Damek and A. Hulanicki, Maximal functions related to subelliptic operators invariant under the action of a solvable group, Studia Math. 101 (1991), 33-68.
  10. [DH3] E. Damek and A. Hulanicki, Boundaries and the Fatou theorem for subelliptic second order operators on solvable Lie groups, Colloq. Math. 68 (1995), 121-140.
  11. [DR] E. Damek and F. Ricci, Harmonic analysis on solvable extensions of H-type groups, J. Geom. Anal. 2 (1992), 213-249.
  12. [Hei] E. Heintze, On homogeneous manifolds of negative curvature, Math. Ann. 211 (1974), 23-34.
  13. [H] A. Hulanicki, Subalgebra of L1(G) associated with laplacian on a Lie group, Colloq. Math. 31 (1974), 259-287.
  14. [M] P. Malliavin, Géometrie Différentielle Stochastique, Sém. Math. Sup. 64, Les Presses de l'Université de Montréal, Montréal, 1977.
  15. [N] E. Nelson, Analytic vectors, Ann. of Math. 70 (1959), 572-615.
  16. [P] I. I. Pyatetskiĭ-Shapiro, Geometry and classification of homogeneous bounded domains in ℂn, Uspekhi Mat. Nauk 20 (2) (1965), 3-51 (in Russian); English transl.: Russian Math. Surveys 20 (1966), 1-48.
  17. [R] A. Raugi, Fonctions harmoniques sur les groupes localement compacts à base dénombrable, Bull. Soc. Math. France Mém. 54 (1977), 5-118.
  18. [Ro] D. W. Robinson, Elliptic Operators and Lie Groups, Oxford Math. Monographs, Clarendon Press, 1991.
  19. [So] J. Sołowiej, Fatou theorem - a negative result, Colloq. Math. 67 (1995), 131-145.
  20. [S] D. Stroock, Lectures on Stochastic Analysis: Diffusion Theory, Cambridge Univ. Press, 1987.
  21. [SV] D. Stroock and S. R. Varadhan, Multidimensional Diffusion Processes, Springer, 1979.
  22. [T] J. C. Taylor, Skew products, regular conditional probabilities and stochastic differential equations: a technical remark, in: Séminaire de Probabilités XXVI, Lecture Notes in Math. 1526, Springer, 1992, 113-126.
  23. [U] K. Urbanik, Functionals of transient stochastic processes with independent increments, Studia Math. 103 (1992), 299-314.
  24. [V1] N. T. Varopoulos, Diffusion on Lie groups (I), Canad. J. Math. 46 (1994), 1073-1093.
  25. [V2] N. T. Varopoulos, The heat kernel on Lie groups, Rev. Mat. Iberoamericana 12 (1996), 147-186.
  26. [V3] N. T. Varopoulos, L. Saloff-Coste and T. Coulhon, Analysis and Geometry on Groups, Cambridge Tracts in Math. 100, Cambridge Univ. Press, 1992.
  27. [Vi] E. B. Vinberg, The theory of convex homogeneous cones, Trans. Moscow Math. Soc. 12 (1963), 340-403.
Studia Mathematica
1997/126/2
Export to PBN, Opens in new tab
Pages:
115-148
Main language of publication
English
Received
1996-05-27
Accepted
1997-03-27
Published
1997
Exact and natural sciences