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