D-harmonic distributions and global hypoellipticity on nilmanifolds

Jacek M. Cygan and Leonard Frederick Richardson

Vol. 147 (1991), No. 1, 29–46

DOI: 10.2140/pjm.1991.147.29

Abstract

Let M = Γ∖N be a compact nilmanifold. A system of differential operators D₁,…,D_k on M is globally hypoelliptic (GH) if when D₁f = g₁,…,D_kf = g_k with f ∈𝒟^′(M), g₁,…,g_k ∈ C^∞(M) then f ∈ C^∞(M). Let X₁,…,X_k be real vector fields on M induced by the Lie algebra 𝒩 of N. We study the relationships between (GH) of the system X₁,…,X_k on M, (GH) of the operator D = X₁² + ⋯ + X_k², the constancy of D-harmonic distributions on M, and related algebraic conditions on X₁,…,X_k ∈𝒩.

Mathematical Subject Classification 2000

Primary: 22E30

Secondary: 22E25, 22E40, 35H05

Milestones

Received: 14 February 1989

Revised: 13 October 1989

Published: 1 January 1991