Let be a Lie group of polynomial volume growth, with Lie algebra . Consider a second-order, right-invariant, subelliptic differential operator on , and the associated semigroup . We identify an ideal of such that satisfies global regularity estimates for spatial derivatives of all orders, when the derivatives are taken in the direction of . The regularity is expressed as estimates for derivatives of the semigroup, and as Gaussian bounds for derivatives of the heat kernel. We obtain the boundedness in , , of some associated Riesz transform operators. Finally, we show that is the largest ideal of for which the regularity results hold. Various algebraic characterizations of are given. In particular, where is the nilradical of and is the largest semisimple ideal of . Additional features of this article include an exposition of the structure theory for 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. 929–996DOI 10.4171/RMI/441