{"abstract":[{"lang":"eng","text":"<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\">\r\n                <mml:msup>\r\n                  <mml:mi>L</mml:mi>\r\n                  <mml:mn>1</mml:mn>\r\n                </mml:msup>\r\n              </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>"}],"status":"public","date_created":"2024-04-17T13:17:37Z","doi":"10.1007/s11118-023-10109-1","publisher":"Springer Science and Business Media LLC","language":[{"iso":"eng"}],"_id":"53540","title":"Large-Time Behavior of Two Families of Operators Related to the Fractional Laplacian on Certain Riemannian Manifolds","date_updated":"2024-04-17T13:19:59Z","publication":"Potential Analysis","type":"journal_article","citation":{"ama":"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>","bibtex":"@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} }","mla":"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>.","ieee":"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>.","short":"E. Papageorgiou, Potential Analysis (2023).","apa":"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>","chicago":"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>."},"keyword":["Analysis"],"user_id":"100325","year":"2023","author":[{"full_name":"Papageorgiou, Efthymia","last_name":"Papageorgiou","first_name":"Efthymia","id":"100325"}],"publication_identifier":{"issn":["0926-2601","1572-929X"]},"department":[{"_id":"555"}],"publication_status":"published"}