Some properties of sub-Laplaceans

2018/09/04 by Nicola Garofalo, Garofalo, Nicola · 1 citation
Computer Science · Mathematics · #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #FOS: Mathematics #Nonlinear Partial Differential Equations #Numerical methods in inverse problems

paper · pdf · doi:10.48550/arxiv.1809.01242

openalex publication_date 2018/09/04 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In this note I present some properties of sub-Laplaceans associated with a collection of smooth vector fields satisfying Hörmander'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.

Citations

• Generalized Bochner formulas and Ricci lower bounds for sub-Riemannian manifolds of rank two

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• Multiple sign-changing solutions for semilinear subelliptic Dirichlet problem

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