标题： Hodge拉普拉斯算子在Heisenberg群上的分析 显示英文标题

标题： Analysis of the Hodge Laplacian on the Heisenberg group

摘要： 我们考虑在Heisenberg群$H_n$上带有左不变和U(n)不变的黎曼度量的Hodge拉普拉斯算子$\Delta$。 对于$0\le k\le 2n+1$，令$\Delta_k$表示限制在$k$-形式上的Hodge拉普拉斯算子。 我们第一个主要结果表明，$L^2\Lambda^k(H_n)$分解为有限多个相互正交的子空间$\V_\nu$，具有以下性质：{项目符号}$\dom \Delta_k$沿着$\V_\nu$分解为$\sum_\nu(\dom\Delta_k\cap \V_\nu)$；$\Delta_k:(\dom\Delta_k\cap \V_\nu)\longrightarrow \V_\nu$对于每个$\nu$；对于每个$\nu$，存在一个希尔伯特空间$\cH_\nu$的$L^2$-截面，该截面是一个在$H_n$上的 U(n)-齐次向量丛，使得$\Delta_k$在$\V_\nu$上的限制与一个显式标量算子酉等价。 {项目符号} 接下来，我们考虑$L^p\Lambda^k$，$1<p<\infty$，并证明同样的分解成立。 更准确地说，我们证明了：{项目符号}的 Riesz 变换$d\Delta_k^{-\half}$是$L^p$有界的；到$\cV_\nu$的正交投影从$(L^2\cap L^p)\Lambda^k$扩展为从$L^p\Lambda^k$到$L^p$闭包 \$\cV _ 显示英文摘要

摘要： 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_