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Uniform Gaussian estimates of the fundamental solutions for heat operators on Carnot groups. (English) Zbl 1036.35061
Let $$G= (\mathbb{R}^N,0)$$ be a Carnot group and denote by $$\Delta_G= \sum_{1\leq j\leq m} X^2_j$$ its canonical sub-Laplacian. Given a positive definite symmetric matrix $$A= (a_{ij})_{1\leq i,j\leq m}$$, let us consider the following heat-type operator on $$\mathbb{R}^{N+1}$$ $H_A= \sum_{1\leq i,j\leq m} a_{ij} X_i X_j- \partial_t.$ For a fixed $$\lambda\geq 1$$ one denotes by $$M_\Delta$$ the set of the symmetric matrices $$A$$ such that $$\Lambda^{-1}|\xi|^2\leq \langle A\xi,\xi\rangle\leq \Lambda|\xi|^2$$, $$\xi\in \mathbb{R}^m$$.
In this paper, the authors are concerned with existence, qualitative properties and uniform Gaussian estimates for the global fundamental solutions $$\Gamma_A$$ for $$H_A$$, with $$A\in M_\Lambda$$. They also obtain existence and uniqueness theorems of Tikhonov type for Cauchy problems related to $$H_A$$. The proofs are simple and direct, relying on few basic tools such as invariant Harnack inequalities and maximum principles.

##### MSC:
 35H20 Subelliptic equations 35A08 Fundamental solutions to PDEs 43A80 Analysis on other specific Lie groups
##### Keywords:
invariant Harnack inequalities; maximum principles