Abstract
AbstractThis work deals with the extension problem for the fractional Laplacian on Riemannian symmetric spaces G/K of noncompact type and of general rank, which gives rise to a family of convolution operators, including the Poisson operator. More precisely, motivated by Euclidean results for the Poisson semigroup, we study the long-time asymptotic behavior of solutions to the extension problem for $$L^1$$
L
1
initial data. In the case of the Laplace–Beltrami operator, we show that if the initial data are bi-K-invariant, then the solution to the extension problem behaves asymptotically as the mass times the fundamental solution, but this convergence may break down in the non-bi-K-invariant case. In the second part, we investigate the long-time asymptotic behavior of the extension problem associated with the so-called distinguished Laplacian on G/K. In this case, we observe phenomena which are similar to the Euclidean setting for the Poisson semigroup, such as $$L^1$$
L
1
asymptotic convergence without the assumption of bi-K-invariance.
Publisher
Springer Science and Business Media LLC
Reference31 articles.
1. L. Abadias and E. Alvarez, Asymptotic behavior for the discrete in time heat equation, Mathematics 10 (2022).
2. L. Abadias, J. González-Camus, P.J. Miana, and J.C. Pozo, Large time behaviour for the heat equation on $${\mathbb{Z}}$$, moments and decay rates, J. Math. Anal. Appl. 500 (2021).
3. I. Alvarez-Romero, B. Barrios, and J.J. Betancor, Pointwise convergence of the heat and subordinates of the heat semigroups associated with the Laplace operator on homogeneous trees and two weighted $$L^p$$ maximal inequalities (2022), arXiv:2202.11210.
4. J.-P. Anker, Sharp estimates for some functions of the Laplacian on noncompact symmetric spaces, Duke Math. J. 64 (1992), 257–297.
5. J.-P. Anker and L. Ji, Heat kernel and Green function estimates on noncompact symmetric spaces, Geom. Funct. Anal. 9 (1999), 1035–1091.