An index theorem and hypoellipticity on nilpotent Lie groups

Kenneth Guy Miller

Vol. 99 (1982), No. 2, 419–426

DOI: 10.2140/pjm.1982.99.419

Abstract

Extending results of Grushin we determine the index of p(x,D) where p(x,ξ) is a polynomial homogeneous with respect to some family of dilations on R^2d and p(x,ξ)≠0 if (x,ξ)≠(0,0). In general these operators are not elliptic. If G is a step two nilpotent Lie group and P is a left invariant differential operator on G which is homogeneous with respect to some family of dilations, we apply this index theorem to prove that P is hypoelliptic if and only if P^∗ is hypoelliptic. This extends a result of Helffer and Nourrigat.

Mathematical Subject Classification 2000

Primary: 22E25

Secondary: 58G05, 35H05

Milestones

Received: 20 November 1980

Published: 1 April 1982