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Harmonic analysis and subriemannian geometry on Heisenberg groups. (English) Zbl 1020.35021
The article is based on lectures for non-specialists and provides a clear and excellent exposition of fundamental principles governing the theory of subelliptic operators. It involves a large spectrum of topics: from elementary remarks on convolutions with singular integral operators, through the Laguerre calculus generalizing the common Fourier transformation, up to the sub-Riemannian geometry necessary for the construction of a parametrix. Both simple illustrative examples and rather advanced results are presented.
The article concerns the operators $$\Delta= X^2_1+\cdots+ X^2_m$$, where $$X_j= \partial/\partial x_j+ 2a_j x_{j+m}\partial/\partial t$$ $$(j= 1,\dots, m)$$ are infinitesimal operators of the Heisenberg group $$\mathbb{H}_m$$ on the space $$\mathbb{R}^{2m+1}$$ of variables $$x_1,\dots, x_{2m},t$$. It involves the harmonic analysis on $$\mathbb{H}_m$$ (multiplicative symbolic calculi for left-invariant convolution operators, sub-elliptic estimates for $$\Delta$$, the heat kernel) and geometry (geodesics, Hamilton-Jacobi equation).

##### MSC:
 35H20 Subelliptic equations 53C17 Sub-Riemannian geometry 22E25 Nilpotent and solvable Lie groups
##### Keywords:
Laguerre calculus