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

2023/08/02 by Papageorgiou, Effie · 1 citation
#22E30 #26A33 #35B40 #58J47 #Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.2308.01366

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 L¹ 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 L¹ asymptotic convergence without the assumption of bi-K-invariance.

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