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The intrinsic hypoelliptic Laplacian and its heat kernel on unimodular Lie groups. (English) Zbl 1165.58012

The authors present an invariant definition of the hypoelliptic Laplacian on sub-Riemannian structures with constant growth vector using the Popp volume form introduced by R. Montgomery [A tour of subriemannian geometries, their geodesics and applications. Mathematical Surveys and Monographs 91. Providence, RI: American Mathematical Society (AMS). (2002; Zbl 1044.53022)]. This definition generalizes the one of the Laplace-Beltrami operator in Riemannian geometry. In the case of left-invariant problems on unimodular Lie groups it coincides with the usual sum of squares.
The method, first used by A. Hulanicki [Stud. Math. 56, 165–173 (1976; Zbl 0336.22007)] on the Heisenberg group, is extended to compute explicitly the kernel of the hypoelliptic heat equation on any unimodular Lie group of type I. The main tool is the noncommutative Fourier transform. Some relevant cases are also studied sich as \(SU(2),\) \(SO(3),\) \(SL(2)\) (with the metrics inherited by the Killing form), and the group \(SE(2)\) of rototranslations of the plane.

MSC:

58J35 Heat and other parabolic equation methods for PDEs on manifolds
43A80 Analysis on other specific Lie groups
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