In this note I present some properties of sub-Laplaceans associated with a collection of smooth vector fields satisfying Hormander's finite rank assumption. One notable aspect of this paper is the development of the fractional powers of sub-Laplaceans as Dirichlet-to-Neumann maps of an extension problem inspired to the famous 2007 work of Caffarelli and Silvestre for the standard Laplacean. A key tool is an extension problem for the fractional heat equation for which I compute the relevant Poisson kernel. I then use the latter to: (1) find the Poisson kernel for the time-independent case; and (2) solve the extension problem.
Published September 15, 2018.
Math Subject Classifications: 35C15, 35K05, 35J70.
Key Words: Sub-Laplaceans; mean-value formulas; fractional powers; extension problem.
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| Nicola Garofalo |
Dipartimento d'Ingegneria Civile e Ambientale (DICEA)
Università di Padova Via Marzolo, 9
35131 Padova, Italy
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