Revista Matemática Iberoamericana


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Volume 21, Issue 3, 2005, pp. 929–996
DOI: 10.4171/RMI/441

Published online: 2005-12-31

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

Nick Dungey[1]

(1) Macquarie University, Sydney, Australia

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

Keywords: Lie group, subelliptic operator, heat kernel, Riesz transform, regularity estimates

Dungey Nick: High order regularity for subelliptic operators on Lie groups of polynomial growth. Rev. Mat. Iberoamericana 21 (2005), 929-996. doi: 10.4171/RMI/441