Analysis of the Hodge Laplacian on the Heisenberg group

2012/06/20 by Detlef Müller, Müller, Detlef, Marco M. Peloso +3
Mathematics · #FOS: Mathematics #Functional Analysis (math.FA) #math.FA

paper · pdf · doi:10.48550/arxiv.1206.4540

arxiv created 2012/06/20 · arxiv updated 2012/06/21

Abstract

We consider the Hodge Laplacian Δ on the Heisenberg group H_n, endowed with a left-invariant and U(n)-invariant Riemannian metric. For 0≤ k≤ 2n+1, let Δ_k denote the Hodge Laplacian restricted to k-forms. Our first main result shows that L²Λ^k(H_n) decomposes into finitely many mutually orthogonal subspaces \V_ν with the properties: itemize \dom Δ_k splits along the \V_ν's as ∑_ν(\domΔ_k∩ \V_ν); Δ_k:(\domΔ_k∩ \V_ν)\longrightarrow \V_ν for every ν; for each ν, there is a Hilbert space \cH_ν of L²-sections of a U(n)-homogeneous vector bundle over H_n such that the restriction of Δ_k to \V_ν is unitarily equivalent to an explicit scalar operator. itemize Next, we consider L^pΛ^k, 1<p<∞, and prove that the same kind of decomposition holds true. More precisely we show that: itemize the Riesz transforms dΔ_k^-\half are L^p-bounded; the orthogonal projection onto \cV_ν extends from (L²∩ L^p)Λ^k to a bounded operator from L^pΛ^k to the the L^p-closure \cV_

Related

• L^p-spectral multipliers for the Hodge Laplacian acting on 1-forms on the Heisenberg group
• The Laplacian on p-forms on the Heisenberg group
• Dirichlet problem for Schrödinger operators on Heisenberg groups
• Hardy spaces associated with Schrodinger operators on the Heisenberg group
• Calderón-Hardy type spaces and the Heisenberg sub-Laplacian