Spectral Multipliers on Lie Groups of Polynomial Growth

Abstract

Let $L$ be a left invariant sub-Laplacian on a connected Lie group $G$ of polynomial volume growth, and let $\{ {E_\lambda },\lambda \geqslant 0\}$ be the spectral resolution of $L$ and $m$ a bounded Borel measurable function on $[0,\infty )$. In this article we give a sufficient condition on $m$ for the operator $m(L) = \smallint _0^\infty m(\lambda )d{E_\lambda }$ to extend to an operator bounded on ${L^p}(G),\;1 < p < \infty$, and also from ${L^1}(G)$ to weak-${L^1}(G)$.