Analysis of the Hodge Laplacian on the Heisenberg group

Abstract

We consider the Hodge Laplacian $\Delta$ on the Heisenberg group $H_n$, endowed with a left-invariant and U(n)-invariant Riemannian metric. For $0\le k\le 2n+1$, let ${\Delta }_k$ denote the Hodge Laplacian restricted to $k$-forms. Our first main result shows that $L^2{\Lambda }^k(H_n)$ decomposes into finitely many mutually orthogonal subspaces $\V_ν$ with the properties: {itemize} $\dom Δ_k$ splits along the $\V_ν$'s as $\sum_ν(\domΔ_k\cap \V_ν)$; $Δ_k:(\domΔ_k\cap \V_ν)\longrightarrow \V_ν$ for every $\nu$; for each $\nu$, there is a Hilbert space $\cH_ν$ of $L^2$-sections of a U(n)-homogeneous vector bundle over $H_n$ such that the restriction of ${\Delta }_k$ to $\V_ν$ is unitarily equivalent to an explicit scalar operator. {itemize} Next, we consider $L^p{\Lambda }^k$, $1<p<\infty$, 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 $(L^2\cap L^p){\Lambda }^k$ to a bounded operator from $L^p{\Lambda }^k$ to the the $L^p$-closure $\cV_