用整体提升方法证明齐次Hörmander算子整体基本解的存在性
The existence of a global fundamental solution for homogeneous Hörmander operators via a global lifting method
摘要：
本文证明了在$\mathbb{R}^n$上的任何Hörmander算子$\mathcal{L}=\sum_{i=1}^m X_i^2$的整体基本解$Γ(x;y)$（带极点$x$）的存在性，其中$Γ(x;y)$是$δ$ -齐次$2$ .利用Folland在[On the Rothschild-Stein lifting theorem，Comm. PDE，1977]中证明的齐次算子的整体提升方法，存在一个Carnot群$\mathbb{G}$和一个多项式满射映射$π:\mathbb{G}\to \mathbb{R}^n$，使得$\mathcal{L}$与$\mathbb{G}$上的一个次拉普拉斯算子$\mathcal{L}_{\mathbb{G}}$相关.我们表明，它总是可以执行一个（全球）的变化$\mathbb{G}$变量，使提升映射$π$成为$\mathbb{R}^n$上的投影$\mathbb{G}\equiv \mathbb{R}^n\times\mathbb{R}^p$。如果$Γ_{\mathbb{G}}(x,{x}';y,{y}')$（$x,{x}'\in\mathbb{R}^n$ ; $y,{y}'\in\mathbb{R}^p$）是$\mathcal{L}_{\mathbb{G}}$的基本解，我们证明了$Γ_{\mathbb{G}}(x,0;y,{y}')$总是可积的. ${y}'\in \mathbb{R}^p$的积分是$\mathcal{L}$的基本解。
英文摘要：
We prove the existence of a global fundamental solution $Γ(x;y)$ (with pole $x$) for any Hörmander operator $\mathcal{L}=\sum_{i=1}^m X_i^2$ on $\mathbb{R}^n$ which is $δ$-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 $π:\mathbb{G}\to \mathbb{R}^n$ such that $\mathcal{L}$ is $π$-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 $π$ becomes the projection of $\mathbb{G}\equiv \mathbb{R}^n\times\mathbb{R}^p$ onto $\mathbb{R}^n$. If $Γ_{\mathbb{G}}(x,{x}';y,{y}')$ ($x,{x}'\in\mathbb{R}^n$; $y,{y}'\in\mathbb{R}^p$) is the fundamental solution of $\mathcal{L}_{\mathbb{G}}$, we show that $Γ_{\mathbb{G}}(x,0;y,{y}')$ is always integrable w.r.t. ${y}'\in \mathbb{R}^p$, and its integral is a fundamental solution for $\mathcal{L}$.
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