Abstract
This paper is concerned with the study of Riesz means of the eigenvalues of the sub-Laplacian
A(Ω)=−X^21−X^22,
in the Heisenberg group H1 with Dirichlet boundary conditions on bounded domains Ω of ℝ3 (X1,X2 form a basis for the associated Heisenberg Lie algebra). The spectrum of the sub-Laplacian is purely discrete, and Hansson and Laptev proved an estimate for Tr(A(Ω)−λ)− in terms of the measure of Ω and λ3.
The authors of this article improve this estimate and obtain an inequality with a sharp leading term and an additional lower-order term. The method that they employ does not rely on a Hardy inequality involving the distance to the boundary; instead they exploit the properties of the Carnot-Carathéodory metric associated to the Heisenberg setting.
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