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Fundamental solutions for non-divergence form operators on stratified groups. (English) Zbl 1072.35011
The aim of the paper is to provide a fundamental solution $$\Gamma$$ for operators of the form $H:= \sum^m_{i,j=1} a_{ij}(x,t) X_i X_j-\partial_t\qquad ((x,t)\in \mathbb{R}^{N+1}),$ where the $$X_i$$ generate a Lie algebra with an additional homogeneous structure and the Hölder continuous functions $$a_{ij}$$ form a matrix which belongs to the set $$M_\Lambda$$ of symmetric $$(m,m)$$-matrices $$A$$ with $\Lambda^{-1}|\xi|^2\leq \langle A\xi,\xi\rangle\leq \Lambda|\xi|^2\qquad (\xi\in\mathbb{R}^m),$ $$\Lambda> 1$$ being fixed once and for all. Constant coefficient operators admit of fundamental solutions that satisfy Gaussian bounds which are independent of $$A\in M_\Lambda$$. This was proved by the authors in [Adv. Differ. Equ. 7, 1153–1192 (2002; Zbl 1036.35061)]. This result yields fundamental solutions $$\Gamma_\varepsilon$$ for operators with suitably regularised functions $$a^\varepsilon_{ij}$$ which approximate the $$a_{ij}$$ uniformly on compact sets. The desired $$\Gamma$$ is then obtained by means of a limiting procedure. After giving appropriate bounds for $$\Gamma$$, a fundamental solution is obtained for the elliptic operator corresponding to $$H$$ by integrating $$\Gamma$$ over the time variable.

##### MSC:
 35A08 Fundamental solutions to PDEs 35H20 Subelliptic equations 43A80 Analysis on other specific Lie groups 35A17 Parametrices in context of PDEs 35J70 Degenerate elliptic equations
Gaussian bounds
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