Analysis of the Hodge Laplacian on the Heisenberg group

TL;DR

This paper analyzes the Hodge Laplacian on the Heisenberg group, revealing a detailed spectral decomposition and establishing $L^p$ boundedness of associated Riesz transforms, advancing understanding of harmonic analysis on this noncommutative space.

Contribution

It provides a spectral decomposition of the Hodge Laplacian on the Heisenberg group and proves $L^p$ boundedness of Riesz transforms, extending analysis to $L^p$ spaces.

Findings

Decomposition of $L^2$ space into orthogonal subspaces with explicit scalar operators.

Extension of the decomposition to $L^p$ spaces for $1<p< olinebreak\infty$.

Boundedness of Riesz transforms $d riangle_k^{-rac{1}{2}}$ on $L^p$ spaces.

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 $\V_\nu$ with the properties: {itemize} $\dom \Delta_k$ splits along the $\V_\nu$'s as $\sum_\nu(\dom\Delta_k\cap \V_\nu)$; $\Delta_k:(\dom\Delta_k\cap \V_\nu)\longrightarrow \V_\nu$ for every $\nu$; for each $\nu$, there is a Hilbert space $\cH_\nu$ of $L^2$-sections of a U(n)-homogeneous vector bundle over $H_n$ such that the restriction of $\Delta_k$ to $\V_\nu$ 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 $d\Delta_k^{-\half}$ are $L^p$-bounded; the orthogonal projection onto $\cV_\nu$ 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_