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

<jats:title>Abstract</jats:title><jats:p>This work deals with the extension problem for the fractional Laplacian on Riemannian symmetric spaces <jats:italic>G</jats:italic>/<jats:italic>K</jats:italic> 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 <jats:inline-formula><jats:alternatives><jats:tex-math>$$L^1$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>L</mml:mi> <mml:mn>1</mml:mn> </mml:msup> </mml:math></jats:alternatives></jats:inline-formula> initial data. In the case of the Laplace–Beltrami operator, we show that if the initial data are bi-<jats:italic>K</jats:italic>-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-<jats:italic>K</jats:italic>-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 <jats:italic>G</jats:italic>/<jats:italic>K</jats:italic>. In this case, we observe phenomena which are similar to the Euclidean setting for the Poisson semigroup, such as <jats:inline-formula><jats:alternatives><jats:tex-math>$$L^1$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>L</mml:mi> <mml:mn>1</mml:mn> </mml:msup> </mml:math></jats:alternatives></jats:inline-formula> asymptotic convergence without the assumption of bi-<jats:italic>K</jats:italic>-invariance.</jats:p>