Some properties of sub-Laplaceans
Analysis of PDEs
2018-09-06 v1
Abstract
In this note I present some properties of sub-Laplaceans associated with a collection of smooth vector fields satisfying H\"ormander's finite rank assumption. One notable aspect of the 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.
Cite
@article{arxiv.1809.01242,
title = {Some properties of sub-Laplaceans},
author = {Nicola Garofalo},
journal= {arXiv preprint arXiv:1809.01242},
year = {2018}
}