Estimates of Riesz sums of Schrödinger operators on Riemannian manifolds and Lie groups. (Estimations des sommes de Riesz d’opérateurs de Schrödinger sur les variétés riemanniennes et les groupes de Lie.) (French. Abridged English version) Zbl 0760.58040
Estimations for Riesz sums of powers of Laplacians on some Riemannian manifolds and sub-Laplacians on Lie groups are given. A connected, complete Riemannian manifold \(M\) of dimension \(n\) with bounded geometry is considered. Then, the Schrödinger operators \(S_ k(t)\) are defined and a theorem showing under what conditions these operators are bounded is proven. This theorem refers to the following two cases: (i) \(M\) is a Riemannian manifold as above or a Lie group \(G\) with polynomial growth of volume; (ii) the Riemannian volume of the geodesic ball \(B_ x(t)\) of center \(x\) and radius \(t\) grows exponentially or \(G\) is not of polynomial growth of volume.
The proof of the theorem is based on the method of finite propagation speed for Laplacians and sub-Laplacians.
The proof of the theorem is based on the method of finite propagation speed for Laplacians and sub-Laplacians.
Reviewer: G.Zet (Iasi)
MSC:
| 58J35 | Heat and other parabolic equation methods for PDEs on manifolds |
| 43A15 | \(L^p\)-spaces and other function spaces on groups, semigroups, etc. |
