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

Abstract

Let $G$ be a Lie group of polynomial volume growth, with Lie algebra $\mathfrak{g}$. Consider a second-order, right-invariant, subelliptic differential operator $H$ on $G$, and the associated semigroup $S_t=e^{-tH}$. We identify an ideal ${\mathfrak{n}}^{\prime }$ of $\mathfrak{g}$ such that $H$ satisfies global regularity estimates for spatial derivatives of all orders, when the derivatives are taken in the direction of ${\mathfrak{n}}^{\prime }$. 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 $L_p$, $1<p<\infty$, of some associated Riesz transform operators. Finally, we show that ${\mathfrak{n}}^{\prime }$ is the largest ideal of $\mathfrak{g}$ for which the regularity results hold. Various algebraic characterizations of ${\mathfrak{n}}^{\prime }$ are given. In particular, ${\mathfrak{n}}^{\prime }=\mathfrak{s}\oplus \mathfrak{n}$ where $\mathfrak{n}$ is the nilradical of $\mathfrak{g}$ and $\mathfrak{s}$ is the largest semisimple ideal of $\mathfrak{g}$. 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.