Asymptotic behavior of solutions to the extension problem for the fractional Laplacian on noncompact symmetric spaces

TL;DR

This paper investigates the long-time asymptotic behavior of solutions to the extension problem for the fractional Laplacian on noncompact symmetric spaces, revealing conditions under which solutions converge to fundamental solutions and highlighting differences based on invariance properties.

Contribution

It extends Euclidean results to noncompact symmetric spaces, analyzing asymptotic behaviors for different Laplacians and invariance conditions, and identifies phenomena similar to Euclidean settings.

Findings

Asymptotic convergence to fundamental solutions for bi-$K$-invariant initial data.

Breakdown of convergence in non bi-$K$-invariant cases for the Laplace-Beltrami operator.

Euclidean-like asymptotic behavior observed for the distinguished Laplacian without invariance assumptions.

Abstract

This 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 $L^1$ initial data. In the case of the Laplace-Beltrami operator, we show that if the initial data is 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 $L^1$ asymptotic convergence without the assumption of bi-$K$-invariance.