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).


35H20 Subelliptic equations
53C17 Sub-Riemannian geometry
22E25 Nilpotent and solvable Lie groups