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 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 \(L^1\) asymptotic convergence without the assumption of bi-K-invariance.

这项工作处理非紧类型和一般秩的黎曼对称空间G / K上分数拉普拉斯的扩展问题，这产生了一系列卷积算子，包括泊松算子。更准确地说，受泊松半群的欧几里得结果的启发，我们研究了\(L^1\)初始数据的可拓问题解的长期渐近行为。在 Laplace–Beltrami 算子的情况下，我们表明，如果初始数据是双K不变的，那么可拓问题的解将渐近地表现为质量乘以基本解，但这种收敛性可能会在非-bi- K不变情况。在第二部分中，我们研究与G / K上所谓的杰出拉普拉斯算子相关的可拓问题的长期渐近行为。在这种情况下，我们观察到类似于泊松半群的欧几里得设置的现象，例如在没有双K不变性假设的情况下\(L^1\)渐近收敛。