Vol. 99, No. 2, 1982

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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