Hostname: page-component-77c89778f8-7drxs Total loading time: 0 Render date: 2024-07-18T03:52:57.180Z Has data issue: false hasContentIssue false
  • English
  • Français

Lp–Lrestimates for the Poisson semigroup on homogeneous trees

Part of: Abstract harmonic analysis Difference and functional equations Difference equations Groups and semigroups of linear operators, their generalizations and applications

Published online by Cambridge University Press:  09 April 2009

Alberto G. Setti
Alberto G. Setti
Affiliation:
Dipartimento di Matematica Università di Milano via Saldini 50 20133 Milano Italy e-mail: setti@dsdipa.mat.unimi.it
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Let be a homogeneous tree of degree at least three. In this paper we investigate for which values of p and r the (σθ)-Poisson semigroup is Lp – Lr,-bounded, and we sharp estimate for the corresponding operator norms.

Keywords

Homogeneous treesPoisson semigroupsspherical analysis.

MSC classification

Secondary: 43A85: Analysis on homogeneous spaces 47D03: Groups and semigroups of linear operators 39A12: Discrete version of topics in analysis
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 64 , Issue 1 , February 1998 , pp. 20 - 32
Copyright
Copyright © Australian Mathematical Society 1998

References

[CS]Clerc, J. L. and Stein, E. M., ‘,Lp multipliers for noncompact symmetric spaces’, Proc. Nat. Acad. Sci. U. S. A. 71 (1974), 3911–3912.CrossRefGoogle ScholarPubMed
[C]Cowling, M. G., ‘The Kunze—Stein phenomenon’, Ann. of Math. 107 (1978), 209–234.CrossRefGoogle Scholar
[CGM]Cowling, M. G., Giulini, S. and Meda, S., ‘Lρ–Lq estimates for functions of the Laplace-Beltrami operator on noncompact symmetric spaces. II’, J. Lie Theory 5 (1995), 1–14.Google Scholar
[CMS1]Cowling, M. G., Meda, S. and Setti, A. G., ‘On spherical analysis on groups of isometries of trees’, preprint.Google Scholar
[CMS2]Cowling, M. G., Meda, S. and Setti, A. G., ‘Estimates for functions of the Laplace operator on homogeneous trees,’, preprint.Google Scholar
[E]Erdélyi, A., Asymptotic expansions (Dover, New York, 1956).Google Scholar
[FTN]Talamanca, A. Figà and Nebbia, C., Harmonic analysis and representation theory for groups acting on homogeneous trees, London Math. Society Lecture Note Ser. 162 (Cambridge University Press, Cambridge, 1991).CrossRefGoogle Scholar
[FTP]Talamanca, A. Figà and Picardello, M., Harmonic analysis on free groups (Marcel Dekker, New York, 1983).Google Scholar
[H]Herz, C. S., ’Sur le phénomène de Kunze-Stein’, C. R. Acad. Sci. Paris (Série A) 271 (1970). 491–493.Google Scholar
[Hö]Hörmander, L., ‘Estimetes for translation invariant operators in Lp spaces’, Acta Math. 104 (1960), 93–140.CrossRefGoogle Scholar
[N]Nebbia, C., ‘Groups of isometries of a tree and the Kunze–Stein phenomenon’, Pacific J. Math. 133 (1988), 141–149.CrossRefGoogle Scholar
[P1]Pytlik, T., ‘Radial function of free groups and a decomposition of the regular representation into irreducible components’, J. Reine Angew. Math. 326 (1981), 124–135.Google Scholar
[P2]Pytlik, T., ‘Radial convolutors on free groups, Studia Math. 78 (1984), 178–183.CrossRefGoogle Scholar
[Y]Yosida, K., Functional analysis (Springer, Berlin, 1980).Google Scholar
[Z]Zygmund, A., Trigonometric series 2nd edition (Cambridge Univ. Press, Cambridge, 1959).Google Scholar