| Abstract: | In this paper first the Laplace operator on a completely nonholonomic Riemannian manifold is defined in an invariant manner and its properties are considered. The method presented for studying it, as well as for the study of other hypoelliptic operators, involves the use of the geometry of nonholonomic manifolds. The nonholonomic metric (Carnot-Carathéodory metric), the Carathéodory measure, and hypoharmonic functions are defined. A theorem on the comparison of the spectra is proved and the connection is established between the bases of eigenfunctions of the ordinary and nonholonomic Laplacians. Conjectures are formulated on the principal term of the spectral asymptotic expansion of the nonholonomic Laplacian, on the structure of the wave fronts, and on the propagation of singularities.Translated from Problemy Matematicheskogo Analiza, No. 11, pp. 96–108, 1990. |
