The existence of a global fundamental solution for homogeneous Hörmander operators via a global lifting method

Abstract

We prove the existence of a global fundamental solution $\Gamma (x;y)$ (with pole $x$) for any Hörmander operator $\mathcal{L}={\sum }_{i=1}^m{X}_i^2$ on ${\mathbb{R}}^n$ which is $\delta$-homogeneous of degree $2$. By means of a global Lifting method for homogeneous operators proved by Folland in [On the Rothschild-Stein lifting theorem, Comm. PDEs, 1977], there exists a Carnot group $\mathbb{G}$ and a polynomial surjective map $\pi :\mathbb{G}\to {\mathbb{R}}^n$ such that $\mathcal{L}$ is $\pi$-related to a sub-Laplacian ${\mathcal{L}}_{\mathbb{G}}$ on $\mathbb{G}$. We show that it is always possible to perform a (global) change of variable on $\mathbb{G}$ such that the lifting map $\pi$ becomes the projection of $\mathbb{G}\equiv {\mathbb{R}}^n\times {\mathbb{R}}^p$ onto ${\mathbb{R}}^n$. If ${\Gamma }_{\mathbb{G}}(x,x^{\prime };y,y^{\prime })$ ($x,x^{\prime }\in {\mathbb{R}}^n$; $y,y^{\prime }\in {\mathbb{R}}^p$) is the fundamental solution of ${\mathcal{L}}_{\mathbb{G}}$, we show that ${\Gamma }_{\mathbb{G}}(x,0;y,y^{\prime })$ is always integrable w.r.t. $y^{\prime }\in {\mathbb{R}}^p$, and its integral is a fundamental solution for $\mathcal{L}$.

Cited by in corpus (2)

• On the Baker-Campbell-Hausdorff Theorem: non-convergence and prolongation issues
• A comparison principle for nonlinear heat Rockland operators on graded groups