Vol. 99, No. 2, 1982

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ISSN: 0030-8730
An index theorem and hypoellipticity on nilpotent Lie groups

Kenneth Guy Miller

Vol. 99 (1982), No. 2, 419–426

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 R2d 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
Received: 20 November 1980
Published: 1 April 1982
Kenneth Guy Miller