Abstract
In this paper we study the chaotic behavior of the heat semigroup generated by the Dunkl-Laplacian on weighted L p spaces. In the case of the heat semigroup associated to the standard Laplacian we obtain a complete picture on the spaces L p(Rn, (φ iρ (x))2 dx) where φ iρ is the Euclidean spherical function. The behavior is very similar to the case of the Laplace–Beltrami operator on non-compact Riemannian symmetric spaces studied by Pramanik and Sarkar.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
F. Astengo and B. Di Blasio, Dynamics of the heat semigroup in Jacobi analysis, Journal of Mathematical Analysis and Applications 391 (2012), 48–56.
J. Cheeger, M. Gromov and M. Taylor, Finite propagation speed, kernel estimates for functions of the Laplace operator, and the geometry of complete Riemannian manifolds, Journal of Differential Geometry 17 (1982), 15–53.
F. Dai and H. Wang, A transference theorem for the Dunkl transform and its applications, Journal of Functional Analysis 258 (2010), 4052–4074.
F. Dai and Y. Xu, Approximation Theory and Harmonic Analysis on Spheres and Balls, Springer Monographs in Mathematics, Springer, New York, 2013.
W. Desch, W. Schappacher and G. F. Webb, Hypercyclic and chaotic semigroups of linear operators, Ergodic Theory and Dynamical Systems 17 (1997), 793–819.
C. F. Dunkl, Differential-difference operators associated to reflection groups, Transactions of the American Mathematical Society 311 (1989), 167–183.
C. F. Dunkl and Y. Xu, Orthogonal Polynomials of Several Variables, Encyclopedia of Mathematics and its Applications, Vol. 81, Cambridge University Press, Cambridge, 2001.
S. Helgason, Topics in Harmonic Analysis on Homogeneous Spaces, Progress in Mathematics, Vol. 13, Birkhäuser, Boston, MA, 1981.
C. Herz and N. Riviere, Estimates for translation invariant operators on spaces with mixed norms, Studia Mathematica 44 (1972), 511–515.
M. F. E. de Jeu, The Dunkl transform, Inventiones Mathematicae 113 (1993), 147–162.
L. Ji and A. Weber, Dynamics of the heat semigroup on symmetric spaces, Ergodic Theory and Dynamical Systems 30 (2010), 457–468.
R. de Laubenfels and H. Emamirad, Chaos for functions of discrete and continuous weighted shift operators, Ergodic Theory and Dynamical Systems 21 (2001), 1411–1427.
A. V. Nikiforov and V. B. Uvarov, Special Functions of Mathematical Physics: A Unified Introduction with Applications, Birkhäuser, Basel, 1988.
M. Pramanik and R. P. Sarkar, Chaotic dynamics of the heat semigroup on Riemannian symmetric spaces, Journal of Functional Analysis 266 (2014), 2867–2909.
M. Rösler, Dunkl operators: theory and applications, in Orthogonal Polynomials and Special Functions (Leuven, 2002), Lecture Notes in Mathematics, Vol. 1817, Springer, Berlin, 2003, pp. 93–135.
M. Rösler, Generalized Hermite polynomials and the heat equation for Dunkl operators, Communications in Mathematical Physics 192 (1998), 519–542.
J. L. Rubio de Francia, Transference principles for radial multipliers, Duke Mathematical Journal 58 (1989), 1–19.
R. P. Sarkar, Chaotic dynamics of the heat semigroup on the Damek–Ricci spaces, Israel Journal of Mathematics 198 (2013), 487–508.
E. M. Stein and G. Weiss, Interpolation of operators with change of measures, Transactions of the American Mathematical Society 87 (1958), 159–172.
M. E. Taylor, L p-estimates on functions of the Laplace operator, Duke Mathematical Journal 58 (1989), 773–793.
S. Thangavelu and Y. Xu, Convolution operator and maximal function for Dunkl transform, Journal d’Analyse Mathématique 97 (2005), 25–55.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Boggarapu, P., Thangavelu, S. On the chaotic behavior of the Dunkl heat semigroup on weighted L p spaces. Isr. J. Math. 217, 57–92 (2017). https://doi.org/10.1007/s11856-017-1438-6
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11856-017-1438-6