# High order regularity for subelliptic operators on Lie groups of polynomial growth

### Nick Dungey

Macquarie University, Sydney, Australia

## Abstract

Let $G$ be a Lie group of polynomial volume growth, with Lie algebra $g$. Consider a second-order, right-invariant, subelliptic differential operator $H$ on $G$, and the associated semigroup $S_{t}=e_{−tH}$. We identify an ideal $n_{′}$ of $g$ such that $H$ satisfies global regularity estimates for spatial derivatives of all orders, when the derivatives are taken in the direction of $n_{′}$. The regularity is expressed as $L_{2}$ estimates for derivatives of the semigroup, and as Gaussian bounds for derivatives of the heat kernel. We obtain the boundedness in $L_{p}$, $1<p<∞$, of some associated Riesz transform operators. Finally, we show that $n_{′}$ is the largest ideal of $g$ for which the regularity results hold. Various algebraic characterizations of $n_{′}$ are given. In particular, $n_{′}=s⊕n$ where $n$ is the nilradical of $g$ and $s$ is the largest semisimple ideal of $g$. Additional features of this article include an exposition of the structure theory for $G$ in Section 2, and a concept of twisted multiplications on Lie groups which includes semidirect products in the Appendix.

## Cite this article

Nick Dungey, High order regularity for subelliptic operators on Lie groups of polynomial growth. Rev. Mat. Iberoam. 21 (2005), no. 3, pp. 929–996

DOI 10.4171/RMI/441