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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
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