article published Efthymia Papageorgiou author 100325 555 department <jats:title>Abstract</jats:title><jats:p>This note is concerned with two families of operators related to the fractional Laplacian, the first arising from the Caffarelli-Silvestre extension problem and the second from the fractional heat equation. They both include the Poisson semigroup. We show that on a complete, connected, and non-compact Riemannian manifold of non-negative Ricci curvature, in both cases, the solution with <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 behaves asymptotically as the mass times the fundamental solution. Similar long-time convergence results remain valid on more general manifolds satisfying the Li-Yau two-sided estimate of the heat kernel. The situation changes drastically on hyperbolic space, and more generally on rank one non-compact symmetric spaces: we show that for the Poisson semigroup, the convergence to the Poisson kernel fails -but remains true under the additional assumption of radial initial data.</jats:p> Springer Science and Business Media LLC2023 eng Analysis 0926-2601 1572-929X10.1007/s11118-023-10109-1 Papageorgiou, Efthymia. “Large-Time Behavior of Two Families of Operators Related to the Fractional Laplacian on Certain Riemannian Manifolds.” <i>Potential Analysis</i>, 2023. <a href="https://doi.org/10.1007/s11118-023-10109-1">https://doi.org/10.1007/s11118-023-10109-1</a>. E. Papageorgiou, Potential Analysis (2023). Papageorgiou, E. (2023). Large-Time Behavior of Two Families of Operators Related to the Fractional Laplacian on Certain Riemannian Manifolds. <i>Potential Analysis</i>. <a href="https://doi.org/10.1007/s11118-023-10109-1">https://doi.org/10.1007/s11118-023-10109-1</a> E. Papageorgiou, “Large-Time Behavior of Two Families of Operators Related to the Fractional Laplacian on Certain Riemannian Manifolds,” <i>Potential Analysis</i>, 2023, doi: <a href="https://doi.org/10.1007/s11118-023-10109-1">10.1007/s11118-023-10109-1</a>. Papageorgiou E. Large-Time Behavior of Two Families of Operators Related to the Fractional Laplacian on Certain Riemannian Manifolds. <i>Potential Analysis</i>. Published online 2023. doi:<a href="https://doi.org/10.1007/s11118-023-10109-1">10.1007/s11118-023-10109-1</a> @article{Papageorgiou_2023, title={Large-Time Behavior of Two Families of Operators Related to the Fractional Laplacian on Certain Riemannian Manifolds}, DOI={<a href="https://doi.org/10.1007/s11118-023-10109-1">10.1007/s11118-023-10109-1</a>}, journal={Potential Analysis}, publisher={Springer Science and Business Media LLC}, author={Papageorgiou, Efthymia}, year={2023} } Papageorgiou, Efthymia. “Large-Time Behavior of Two Families of Operators Related to the Fractional Laplacian on Certain Riemannian Manifolds.” <i>Potential Analysis</i>, Springer Science and Business Media LLC, 2023, doi:<a href="https://doi.org/10.1007/s11118-023-10109-1">10.1007/s11118-023-10109-1</a>. 535402024-04-17T13:17:37Z2026-07-03T12:37:31Z