Opérateurs uniformément sous-elliptiques sur les groupes de Lie. (Uniformly sub-elliptic operators on Lie groups)
Let G be a connected unimodular Lie group G, and let \(X_ 1,...,X_ k\) be left invariant vector fields which generate the Lie algebra of G. Let \(a: G\to {\mathbb{R}}^ k\otimes {\mathbb{R}}^ k\) be a smooth symmetric matrix valued function which satisfies \[ \alpha | \xi |^ 2\leq \sum a_{ij}(x)\xi_ i\xi_ j \leq \alpha^{-1}| \xi |^ 2,\quad x\in G,\quad \xi \in {\mathbb{R}}^ k,\text{ for some } \alpha \in]0,1]. \] Put \({\mathcal L}_{\alpha}=\sum^{k}_{i,j=1}X_ ia_{ij}X_ j\). If G has polynomial volume growth, the authors prove Gaussian estimates for the heat kernel \(e^{t{\mathcal L}\alpha}\). As a consequence they obtain regularity results and Harnack inequalities. Their results extend earlier results of De Giorgi-Nash-Moser-Aronson (for parabolic operators in \({\mathbb{R}}^ n)\) and of Varopoulos (for evolution operators associated to the Kohn Laplacian).
scientific article; zbMATH DE number 4215002
scientific article