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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.

35H20 Subelliptic equations
35A08 Fundamental solutions to PDEs
43A80 Analysis on other specific Lie groups