Spectral and Hardy Inequalities for the Heisenberg Laplacian

In this thesis we consider the first Heisenberg group and study spectral properties of the Dirichlet sub-Laplacian, also known as Heisenberg Laplacian, which is a sum-ofsquares differential operator of left-invariant vector fields on the first Heisenberg group. In particular, we consider the bound for the trace of the eigenvalues which reflects the correct geometrical constant and order of growth in Weyl’s law and improve this inequality by adding an additional negative lower order term. In addition we investigate on a Hardy-type inequality for the gradient of the Heisenberg Laplacian on bounded domains since an application of such inequalities improves the growing order of the additional lower order term. Let 0 < λ1(Ω) ≤ λ2(Ω) ≤ . . . denote the eigenvalues of the Heisenberg Laplacian −∆H := −X 1 −X 2 , X1 := (∂x1 + 1 2 x2∂x3), X2 := (∂x2 − 1 2 x1∂x3) for (x1, x2, x3) ∈ R with Dirichlet boundary conditions on a bounded domain Ω ⊂ R. In this thesis we improve the result in [HL08] by A.M. Hansson and A. Laptev ∑ k∈N (λ− λk(Ω))+ ≤ |Ω| 96 λ, λ ≥ 0. We stress that the geometrical constant and order of growth in λ cannot be improved further. Therefore we add an additional negative lower order term to the right-hand side of that inequality. Such inequalities yield immediately bounds for the eigenvalue sum. In addition we show that the growing order of the additional lower order term in our result can be further improved if there exists a constant c(Ω) > 0 independent of u ∈ C∞ 0 (Ω) such that the following Hardy-type inequality holds 1 c(Ω) ∫ Ω |u(x)| δC(x) dx ≤ ∫ Ω |X1u(x)| + |X2u(x)| dx, u ∈ C∞ 0 (Ω). The Hardy weight δC is the distance function to the boundary of Ω measured with respect to the Carnot-Carathéodory metric generated by the span of X1 and X2. In this thesis we show that for open bounded convex polytopes this inequality holds and give explicit estimates on the constant c(Ω).